A Matrix Ring Description for Cyclic Convolutional Codes

A Matrix Ring Description for Cyclic Co n v olutional Co des Heide Gluesing-Luerssen ∗ , F ai-Lung Tsang † No ve mber 20, 2018 Abstract: In this pap er, w e s tu dy con vol u tional co d es with a sp eciﬁc cyclic structure. By deﬁ nition, these co des are left ideals in a certain skew p olynomial rin g. Using that the skew p olynomial ring is isomorp h ic to a matrix r ing w e can describ e the alg ebr aic parameters of the co d es in a more accessible w ay . W e sho w that th e existence of su c h co des with giv en algebraic parameters can b e reduced to the solv abilit y of a mo diﬁ ed rook problem. I t is our strong belief that t h e rook problem is alwa ys solv able, and w e presen t solutions in particular cases. Keyw ords: Con vol u tional c o des, cyclic c o des, sk ew p olynomial rings, F orney ind ices. MSC (2000): 94B10, 94B15, 16S36 1 In tro d uction Con volutio n al cod es (CC’s, for short) form an imp o r tan t class of error-correcting co d es in engineering p ractice. The mathematical theory of these co des h as b een set oﬀ b y the seminal pap ers of F orney [3] and Massey et al. [15], and the progress ev er since is reﬂected b y , for instance, the b ooks [12, 19] and the article [16]. Sev eral directions ha ve b een pursu ed. In the 1970s, a lot of eﬀort has b e en made to construct p o werful CC’s with the aid of go o d b lo c k co des, see [14, 13]. This id ea has b een resumed in the pap e r s [24, 7]. F u rthermore, the metho ds of linear systems theory h a v e b een utilized in order to gain a deep er mathematical understanding of CC’s. W e refer to the pap ers [15, 21, 23, 10] for further d etails and constr u ctions. A third direction in the theory of CC’s dev elop ed when cod es w ith some add itional algebraic structur e were stud ied. Besides the recen tly in tro d uced classes of Goppa con vo lu tional co des [1 , 17] and group con v olutional codes [2 ], the m ain class o f suc h codes are cyclic conv olutional codes. Cyclic structure for C C’s has b een in v estigated f or th e ﬁrst time in the pap ers [18, 20]. One of t h e crucial observ ations reve aled that CC’s that are in v ariant under the ord inary cyclic shift ha ve degree z er o, that is, they are cyclic blo c k co d es. This insigh t has led to ∗ Universit y of K entuc ky , Department of Mathematics, 715 Patterson Oﬃce T o wer, Lexington, KY 40506- 0027, USA; heidegl@ms.uky .edu † Universit y of Groningen, Department of Mathematics, P . O . Bo x 800, 9700 A V Groningen, The Nether- lands; F.L.Tsa n g@math.rug.nl 1 a more complex notio n of c yclicit y for CC’s whic h can b e summarized as fo llows. C y clic con vo lutional codes (CCC ’s, for s hort) are direct summands of F [ z ] n that are at the same time left ideals in a skew p ol yn omial r ing A [ z ; σ ] , where A = F [ x ] / h x n − 1 i and σ is an F - automorphism of A . During t h e last c oup le of y ears a d etailed algebraic theo r y o f CCC’s has b een dev elop ed in [8, 6]. Among other thin gs it h as b een sho wn that CCC’s are principal left ideals in A [ z ; σ ] and , using a t yp e of Gr¨ obner basis theory , one can compute reduced generator p olynomials from wh ic h all algebraic parameters of the co de can easily b e read oﬀ. The details will b e given later on in Th eorem 3.5. C lasses of CCC’s with go o d error-correcting prop erties ha ve b een presen ted in [5, 9]. In this pap er we w ant to contin ue th e inv estigatio n of CCC ’s. W e restrict ourselv es to a particular cl ass of automorphisms. In that case the skew p olynomial ring A [ z ; σ ] turn s out to b e isomorphic to a matrix ring o ver a comm u tativ e p olynomial ring. Th is allo ws u s to easily construct generator p olynomials of CCC ’s w ith prescrib ed algebraic p arameters. Moreo v er, w e discuss the existence of CCC’s with an y given algebraic parameters and will sho w that it reduces to a particular combinato r ial problem resem bling the classic al ro ok problem. W e can solve particular ins tances of that pr oblem, bu t unfortunately n ot th e general c ase. Ho wev er, we strongly b elieve that the com binatorial problem is s olv able for all p ossible cases. The outline of the pap er is as follo w s. In the n ext section we review imp ortant notions from con vo lu tional co d ing theory and introdu ce th e algebraic framew ork for CCC’s. In Section 3 w e concen trate on a particular class of CCC ’s and establish an isomorphism b et ween the asso ciated sk ew p olynomial rin g and a certain mat r ix ring. W e t r anslate the main notions n eeded for the theory of C C C’s in to the matrix setting. In Section 4 w e construct particular CCC’s and discuss th e existence of CC C’s with give n F orney ind ices. The existence of su ch co d es reduces to a com b in atorial problem follo w ed by a problem of constructing polynomial matrices with certain degree p rop erties. Af ter p resen ting a pro of of the matrix problem w e d iscuss and solv e particular instances of the com b inatorial problem in Section 5. W e close the pap er with a sh ort section illustrating h o w to generalize the r esults to codes t h at are cyclic with respect to a general automorphism. 2 Preliminaries on Cyclic Con v olutional Co des Throughout this p ap er let F b e a ﬁ n ite ﬁeld with q ele ments. A con volutio n al co d e of length n and dimension k is a submod ule C of F [ z ] n ha ving the form C = im G := { uG   u ∈ F [ z ] k } , where G is a basic matrix in F [ z ] k × n , i. e. rank G ( λ ) = k for all λ ∈ F (with F b eing an algebraic closure of F ). W e call suc h a matrix G an encod er , and the n umb er deg ( C ) := deg( G ) := max { deg ( M ) | M is a k -min or of G } is sa id to b e th e degree of the enco der G or of the co de C . Recall that the requirement of G b eing basic (rather th an just ha ving full ro w rank o ver F [ z ]) is equiv alen t to C b eing a direct summand of the mod ule F [ z ] n . Ob v ious ly , tw o basic matrices G, G ′ ∈ F [ z ] k × n satisfy im G = im G ′ if and only if G ′ = U G for some U ∈ GL k ( F [ z ]) : = { V ∈ F [ z ] k × k | det( V ) ∈ F \{ 0 }} . It is we ll kno wn [4, p. 495] 2 that eac h CC admits a minimal enco der . Here a matrix G ∈ F [ z ] k × n is said to b e minimal if the s u m of its ro w degrees equals deg( G ), where the degree of a p olynomial ro w v ector is deﬁned a s the ma xim al degree of its entries. F or detail s see, e . g., [4 , Main Thm.] or [16, Thm. A.2]. Such a minimal enco der is in general not uniqu e; the ro w degrees of a minimal enco der, ho wev er, are, up to ord ering, u niquely determined by the co de and are called the F orn ey indices of the code or o f th e enco der. It follo ws that a CC h as a constan t encod er matrix if and only if the degree is zero. I n that c ase t h e co de is, in a natural wa y , a b lo c k co de. Bey ond these pur ely alge b raic concepts the most imp ortant notion in error-con trol co ding is certainly th e weig ht. F or a p olynomial v ector v = P N j =0 v ( j ) z j ∈ F [ z ] n , where v ( j ) ∈ F n , one deﬁnes its weigh t as wt( v ) = P N j =0 wt( v ( j ) ), with wt( v ( j ) ) denoting the (Hamming) w eight of th e v ector v ( j ) ∈ F n . Just like for blo c k co des the distance of a co de C is deﬁned as d ist( C ) = min { wt( v ) | v ∈ C , v 6 = 0 } . Let us no w turn to the notio n of cyclicit y (for details w e refer to [8, 6]). W e will restrict ourselv es t o cod es where the length n is coprime with the ﬁeld siz e q . F rom the theory of cyclic b lo c k codes reca ll the sta n dard identiﬁcat ion p : F n − → A := F [ x ] / h x n − 1 i , ( v 0 , . . . , v n − 1 ) 7− → n − 1 X i =0 v i x i (2.1) of F n with the ring of p o lyn omials m o dulo x n − 1. E xtending this map coeﬃcient w ise w e can iden tify the p olynomial mod ule F [ z ] n = { P N ν =0 z ν v ν | N ∈ N 0 , v ν ∈ F n } with th e p olynomial ring A [ z ] := n N X ν =0 z ν a ν    N ∈ N 0 , a ν ∈ A o . F ollo win g the theo r y of cyclic block co des one would lik e to d eclare a cod e C ⊆ F [ z ] n cyclic if it is in v arian t und er the cyclic shift acti n g on F [ z ] n , or, equiv alen tly , if its imag e in A [ z ] is an ideal. Ho wev er, it has b een sho wn in v arious ve r sions in [18, Th m. 3.12], [20, Thm. 6], and [8, Prop. 2.7] that ev ery CC with th is prop ert y h as d egree zero. In other w ords, this n otion do es not lead to any co d es other than cyclic blo ck co des. Due to t h is resu lt a more general notion of cycl icity has b een in tro duced and discussed in the pap ers men tioned ab ov e. In order to pr esent this notion w e need the group Aut F ( A ) of all F -automorphism s on A . It is clear that eac h automorphism σ ∈ Aut F ( A ) is un iqu ely determined by the single v alue σ ( x ) ∈ A , but not eve r y choi ce for σ ( x ) d etermines an automorphism on A . Indeed, sin ce x generates the F -algebra A the same has to b e true for σ ( x ), and we obtain for a ∈ A σ ( x ) = a determines an F -automorphism on A  ⇐ ⇒  1 , a, . . . , a n − 1 are linearly ind ep endent o v er F and a n = 1 . (2.2) Fixing an arb itrary auto m orphism σ ∈ Aut F ( A ) we d eﬁne a n ew multiplicatio n on the F [ z ]-mo d ule A [ z ] via az = z σ ( a ) for all a ∈ A. (2.3) Along with asso ciativit y , distributivit y , and the usual m ultiplication insid e A this turns A [ z ] in to a ske w p o lyn omial ring whic h we will denote by A [ z ; σ ]. Notice th at A [ z ; σ ] is 3 non-comm utativ e unless σ is the iden tit y . Moreo ver, the map p fr om (2.1) can b e extended to p : F [ z ] n − → A [ z ; σ ] , N X ν =0 z ν v ν 7− → N X ν =0 z ν p ( v ν ) , (2.4) this wa y yielding an isomorph ism of left F [ z ]-mo dules. No w we declare a subm o dule C ⊆ F [ z ] n to b e σ - cyclic if p ( C ) is a left ideal in A [ z ; σ ]. It is straigh tforward to see that the latter is equiv alen t to the F [ z ]-sub mo dule p ( C ) of A [ z ; σ ] b eing closed u nder left m ultiplication by x . C om binin g this with the d eﬁnition of CC’s w e arrive at the follo win g notion. Deﬁnition 2.1 A subm o dule C ⊆ F [ z ] n is called a σ -cyclic con v olutional co de ( σ -CCC, for short) if C is a direct s ummand of F [ z ] n and p ( C ) is a left ideal in A [ z ; σ ]. In the pap ers [18, 8, 6, 5, 9] the algebraic prop erties of these co des ha ve b e en in vestig ated in detail (the main results are su mmarized in Th eorem 3.5 b elo w) and plent y of CCC ’s, all optimal with resp ec t to their free distance, ha ve b een presen ted. In [5 ] a class of one-dimensional CC C’s has b een constru cted all of w h ic h mem b ers are MDS co des, that is, th ey ha ve the b est distance among all one-dimensional co des with the same length, same degree and ov er any ﬁ nite ﬁeld. This result h as b e en generalized to a class of Reed- Solomon con volutio n al co des in [9]. In [2] the concept of cyclicit y has b e en generalized to group conv olutional co des. Let us close th is section w ith some basic notation. In an y r ing w e will denote left, resp ec- tiv ely righ t, ideals by • h · i , resp ectiv ely h · i • . As u sual, ideals in comm utativ e rings will b e w ritten as h · i . The group of units in a rin g R will b e den oted by R × . 3 The M atrix Ring M In this pap e r w e will restrict our selv es to a more sp eciﬁc class of cyclic co des than those b eing introdu ced in the pr evious section. Ind eed, w e will consid er th e follo wing situation. Let F b e an y ﬁeld w ith q elemen ts and let n ∈ N ≥ 2 b e a d ivisor of q − 1. Th en , using the Chinese Remainder T heorem, the qu otien t ring A ′ := F [ x ] / h x n − 1 i is isomorph ic to the direct p ro du ct A := F × · · · × F | {z } n copies . (3.1) The elements of A will b e denoted as a = [ a 1 , . . . , a n ], where a i ∈ F for i = 1 , . . . , n . Observe that F naturally emb eds into A via f 7− → [ f , . . . , f ], thus A is an F -algebra. This algebra structure is isomorphic to the n atur al F -algebra structur e of A ′ . The stand ard F -b asis of A is giv en by { e 1 , . . . , e n } , wh ere e i = [0 , . . . , 1 , . . . , 0] with the 1 app earing in th e i -th p osition . (3.2) Ob v ious ly , these are just the pr imitiv e idemp ote nts of the ring A , and A = ⊕ n i =1 h e i i as a direct sum of ideals. I t is cle ar that the automorphisms of A are in one-one co r resp ond ence 4 with the p erm u tations of th e primitiv e idemp oten ts. As a consequence, | Aut F ( A ) | = n !, see also [8, Cor. 3.2]. In this pap e r , w e will consider only those automorphisms for whic h the p erm utation is a cycle of length n . It is not hard to see that there are exactly ( n − 1)! automorphisms of that kind. Allo wing a suitable p erm utation of the n copies of F , w e ma y restrict ourselve s to the automorphism σ : A − → A, σ ([ a 1 , . . . , a n ]) = [ a n , a 1 , . . . , a n − 1 ] . (3.3) Then σ ( e i ) = e i +1 for 1 ≤ i ≤ n − 1 and σ ( e n ) = e 1 , and w e h a v e the identitie s σ j ( e i ) = e ( i + j − 1 m od n )+1 for all i = 1 , . . . , n and j ∈ N 0 . (3.4) As in the pr evious section, the automorphism σ giv es rise to a sk ew p olynomia l ring ( A [ z ; σ ] , + , · ) w here, again, th e set { P N ν =0 z ν a ν | N ∈ N 0 , a ν ∈ A } is equip p ed with the usual coord inatewise addition, and where m ultiplication is deﬁned by the rule az = z σ ( a ) for all a ∈ A , see (2.3). As already indicated by the notation ab ov e, coeﬃcien ts of the p olynomials in A [ z ; σ ] are alwa ys m eant to b e the right hand side co eﬃcien ts. Again, th e left F [ z ]-modu le A [ z ; σ ] is isomorph ic to F [ z ] n via the map p from (2.4), and wh ere we use an isomorphism of A w ith the quotien t r in g A ′ . Thus, A [ z ; σ ] giv es us the fr amework f or the class of C CC’s wh ere the length n divides q − 1 and where the un derlying automorph ism induces a cycle of length n on the pr imitiv e idemp oten ts. Example 3.1 L et F = F 4 = { 0 , 1 , α, α 2 } , wh ere α 2 = α + 1, and let n = 3. Then A = F × F × F ∼ = F [ x ] / h x 3 − 1 i = A ′ and | Aut F ( A ) | = 6. The automorphisms σ ∈ Aut F ( A ′ ) are completely determined b y their v alue σ ( x ) assigned to x , see also (2.2). T h ese v alues are given by x, x 2 , αx, α 2 x, αx 2 , α 2 x 2 . The t wo automorphisms inducing cycles of length 3 are giv en b y σ 1 ( x ) = αx an d σ 2 ( x ) = α 2 x . Indeed, using the isomorphism φ : A ′ − → A, f 7− → [ f (1) , f ( α ) , f ( α 2 )] the primitive idemp o tents of A are e 1 = [1 , 0 , 0] = φ ( x 2 + x + 1) , e 2 = [0 , 1 , 0] = φ ( αx 2 + α 2 x + 1), and e 3 = [0 , 0 , 1] = φ ( α 2 x 2 + αx + 1). One easily ve r iﬁes th at σ 2 ( e 1 ) = e 2 , σ 2 ( e 2 ) = e 3 , σ 2 ( e 3 ) = e 1 , th u s σ 2 satisﬁes (3.3) (where, of course, w e identify σ 2 ∈ Aut F ( A ′ ) with φσ 2 φ − 1 ∈ Aut F ( A )). Likewise, using the identiﬁca tion A ′ − → A, f 7− → [ f ( α 2 ) , f ( α ) , f (1)] the automorphism σ 1 satisﬁes (3.3). Notice that σ 2 = σ − 1 1 . In the rest of th is section we w ill sho w ho w the sk ew p olynomia l ring A [ z ; σ ] can b e de- scrib ed as a certain matrix ring, and we will translate v arious p rop erties int o the matrix setting. T h is will lead us to a new wa y of constructing CCC’s with giv en algebraic p a- rameters. Let us consider the rin g F [ t ] n × n of p olynomial matrices in an indeterminate t and d eﬁne the subset M :=  ( m ab ) ∈ F [ t ] n × n   m ab (0) = 0 for all 1 ≤ b < a ≤ n  . Notice that M consists exactly of all matrices where the elemen ts b e low the diagonal are m ultiples of t . It is easy to see th at M is a (n on-comm utativ e) subr ing of F [ t ] n × n . 5 Prop osition 3.2 Let A [ z ; σ ] and M b e as ab o ve. Th en the map ξ : A [ z ; σ ] − → M N X l =0 z nl n − 1 X i =0 z i [ c l i 1 , c l i 2 , . . . , c l in ] 7− → N X l =0 t l         c l 01 c l 12 · · · · · · c l n − 1 ,n tc l n − 1 , 1 c l 02 · · · · · · c l n − 2 ,n . . . . . . . . . . . . tc l 21 . . . . . . c l 1 n tc l 11 tc l 22 · · · tc l n − 1 ,n − 1 c l 0 n         (3.5) is a ring isomorp hism. The identi ﬁ cation of A [ z ; σ ] and M was ﬁrst sho w n in [11], where it h as b een stud ied for the more general situation of sk ew p olynomial rings o ve r arbitrary semisimple rings with a monomorphism σ . This g en eral situatio n has also b e en used in [2] in order to classify certain group con v olutional co des. Our c hoice of the semisimple ring A and the automorphism σ leads to a particularly simple pro o f wh ic h we w ould lik e to brieﬂy pr esen t. Pr o of: It is ob vious that eac h p olynomial in A [ z ; σ ] has a unique repr esen tation as on the left hand side of (3.5). Th us the map ξ is well- d eﬁned. Mo r eo v er, it is ob vious that ξ is bijectiv e and add itiv e, and it remains to sho w that it is multiplica tive. In order to do so, w e ﬁrstly obs erv e that ξ ( α ) = diag( α 1 , . . . , α n ) f or any α = [ α 1 , . . . , α n ] ∈ A , and seco n dly , that ξ ( z nl + i ) = t l  0 I n − i tI i 0  =  0 I n − 1 t 0  nl + i for all l ∈ N 0 and i = 0 , . . . , n − 1 . No w the iden tities ξ ( αβ ) = ξ ( α ) ξ ( β ) , ξ ( z ν + µ ) = ξ ( z ν ) ξ ( z µ ), and ξ ( z ν α ) = ξ ( z ν ) ξ ( α ), where ν, µ ∈ N 0 and α, β ∈ A , are ob v ious , and one easily v eriﬁes ξ ( α ) ξ ( z ) = ξ ( z ) ξ ( σ ( α )) for all α ∈ A . Using the additivit y of ξ w e conclud e that ξ is indeed m u ltiplicativ e, hence a rin g isomorphism.  F or later pur p oses it will b e handy to hav e an explicit formula for the entries of ξ ( g ). F or g := N X l =0 z nl n − 1 X i =0 z i [ c l i 1 , c l i 2 , . . . , c l in ] (3.6) and ξ ( g ) = ( m ab ) one computes m ab = N X l =0 t l +sgn( a − b ) c l b − a +sgn( a − b ) n,b for 1 ≤ a, b ≤ n , (3.7) where sgn( x ) :=  1 if x > 0 , 0 else . It is clear that the su bring ξ ( F [ z ]) of M is giv en by the set of matrices 6 N X l =0 t l         c l 0 c l 1 · · · · · · c l n − 1 tc l n − 1 c l 0 · · · · · · c l n − 2 . . . . . . . . . . . . tc l 2 . . . . . . c l 1 tc l 1 tc l 2 · · · tc l n − 1 c l 0         where N ∈ N 0 , c l i ∈ F , (3.8) and, deﬁning f · M := ξ ( f ) M for f ∈ F [ z ] and M ∈ M , w e m a y imp ose a left F [ z ]-modu le structure on M making it isomorphic to A [ z ; σ ] as left F [ z ]-modules. Due to the form of ξ ( f ) as giv en in (3.8) the th us obtained mo d ule structur e is of course not id en tical to the canonical F [ t ]-mo dule structure of M . In the s equel we will translate v arious prop ertie s of p olynomials in A [ z ; σ ] in to matrix prop erties in M . First of all we will sho w ho w to identify the u n its in M . This will pla y an imp orta nt role later on w hen discussin g left ideals th at are direct su mmands. Prop osition 3.3 Let M × b e the group of un its of M . Then M × = M ∩ GL n ( F [ t ]) . In other words, M ∈ M is a un it in the ring M if and only if det( M ) ∈ F × := F \ { 0 } . As a consequence, left- (or r igh t-) in vertible elements in M or in A [ z ; σ ] are un its. Pr o of: The inclus ion “ ⊆ ” is clear. F or the inclusion “ ⊇ ” let M ∈ M ∩ GL n ( F [ t ]). T h en there exists a matrix N ∈ GL n ( F [ t ]) suc h that M N = I . Substituting t = 0 we obtain M (0) N (0) = I , th u s N (0) = M (0) − 1 . Since M ∈ M the matrix M ( 0) is u pp er triangular. But th en the same is true for N (0), sh o wing that N ∈ M .  Let us no w turn to prop e r ties of th e sk ew p o lyn omial ring A [ z ; σ ] that follo w from the semi-simplicit y of A . Since the idemp otents e 1 , . . . , e n are pairwise orthogonal and satisfy e 1 + . . . + e n = 1 w e ha ve A [ z ; σ ] = h e 1 i • ⊕ · · · ⊕ h e n i • = • h e 1 i ⊕ · · · ⊕ • h e n i . As a consequence, eac h element g ∈ A [ z ; σ ] h as a un ique decomp o s ition g = g (1) + . . . + g ( n ) , where g ( a ) := e a g . (3.9) The follo win g notions will play a central role. Reca ll that co eﬃcien ts of sk ew p o lyn omials are alwa ys mean t to b e right hand s id e co eﬃcien ts. Deﬁnition 3.4 Let g ∈ A [ z ; σ ]. (a) F or a = 1 , . . . , n w e call g ( a ) := e a g the a -th c omp onent of g . (b) Th e supp ort of g is deﬁn ed as T g :=  a | g ( a ) 6 = 0  . (c) W e call g delay-fr e e if T g = T g 0 where g 0 ∈ A is the constan t term of the p olynomial g . (d) Th e p olynomial g is said to b e semi- r e duc e d if the leading coeﬃcien ts of the comp o- nen ts g ( a ) , a ∈ T g , lie in pairwise d iﬀeren t ideals h e i a i . (e) The p olynomial g is called r e duc e d if no leading term of an y comp onent of g is a r igh t divisor of a term of any other comp onen t of g . 7 In order to commen t on these notions let us ha ve a closer lo ok at the comp onen ts of a p oly- nomial. Using (3.4) one obtains that, for instance, the ﬁrst comp o n en t of a p olynomial g is of the form g (1) = [ c 0 , 0 , . . . , 0] + z [0 , c 1 , 0 , . . . , 0] + z 2 [0 , 0 , c 2 , 0 , . . . , 0] + · · · = c 0 e 1 + z c 1 e 2 + z 2 c 2 e 3 + · · · for some c j ∈ F . In general we derive from (3.4) g ( a ) = N a X j =0 z j c a,j e ( a + j − 1 mod n )+1 for some N a ∈ N 0 and c a,j ∈ F (3.10) for a = 1 , . . . , n . In p articular, th e co eﬃcien ts of the comp o n en ts are F -m u ltiples of primitiv e idemp otents. As a consequence, g is s emi-reduced if and only if no leading term of any comp onen t of g is a r igh t divisor of the leading term of any other comp onent of g . Obviously , red u cedness implies semi-reducedness, and a p olynomial consisting of one comp onent is alw ays r ed uced. The concept of reducedness has b e en in tro d uced f or the ske w p olynomial ring A [ z ; σ ] in [8, Def. 4.9]. I t p ro ve d to b e v ery useful in the theory of CCC’s. In particular, it h as led to the follo wing results concerning minimal en co der matrices and F orn ey in d ices of CCC ’s, see [8, Thm. 4.5, Thm. 4.15, Prop. 7.10, Th m. 7.13(b)]. Theorem 3.5 As b efore let n | ( q − 1) . Let σ b e an y automorphism in Au t F ( A ) and consider the iden tiﬁcation p of F [ z ] n with the sk ew polynomial ring A [ z ; σ ] as giv en in (2.4) and wh ere we iden tify A with the quotien t ring F [ x ] / h x n − 1 i . (1) F or ev ery g ∈ A [ z ; σ ] there exists a unit u ∈ A [ z ; σ ] × suc h that ug is reduced. (2) Let C ⊆ F [ z ] n b e a σ -CCC . Then there exists a reduced and dela y-free p olynomia l g ∈ A [ z ; σ ] such that p ( C ) = • h g i := { f g | f ∈ A [ z ; σ ] } . In particular, the left ideal p ( C ) is principal. Mo r eo v er, the p olynomial g is u n ique up to left m u ltiplication b y u nits in A . (3) Let g ∈ A [ z ; σ ] b e a r ed uced p olynomial . Then p − 1 ( • h g i ) ⊆ F [ z ] n is a dir ect summand of F [ z ] n if and only if g = P l ∈ T g u ( l ) for some u nit u ∈ A [ z ; σ ] . T h at is, the σ -CCC’s are obtained b y taking an y unit u in A [ z ; σ ] and choosing an y co llection of comp onents of u that forms a r educed p olynomial. (4) Let g ∈ A [ z ; σ ] b e a reduced p olynomial with su p p ort T g = { i 1 , . . . , i k } . T hen C := p − 1 ( • h g i ) = im G = { uG | u ∈ F [ z ] k } , wh ere G :=    p − 1 ( g ( i 1 ) ) . . . p − 1 ( g ( i k ) )    ∈ F [ z ] k × n . (3.11) F u rthermore, if C is a direct su mmand of F [ z ] n then G is a minimal encoder ma- trix of C . As a consequence, C is a σ -CCC of dimension k with F orney indices deg g ( i 1 ) , . . . , deg g ( i k ) and d egree δ := P k l =1 deg g ( i l ) . Let us n o w return to the s ituation where σ is as in (3.3). Throughout this pap er semi- reduced p olynomial s will b e m uc h more handy than reduced ones, see Pr op osition 3.10(4) 8 b elo w. F ortunately , it can easily b e conﬁrmed that the w eake r notion of semi-reducedness is suﬃcient for the r esu lts ab o ve to b e true. W e conﬁne ourselve s to pr esen ting the follo win g details. Remark 3.6 The results of Th eorem 3.5(1) – (4), except for the u niqueness result in (2), are tru e for semi-red u ced p olynomials as well. In order to see th is, one has to conﬁrm that, ﬁ rstly , all results of [8, Section 4], in particular Pr op osition 4.10, C orollary 4.13, and Lemma 4.14, with the exception of the uniqu en ess result in Theorem 4.15, r emain true for semi-reduced p olynomials. S econdly one can easily see that Th eorem 7.8, Prop osition 7.10, Corollary 7.11, Theorem 7.13, and Corollary 7.15 of [8 , Section 7] remain true if one replaces redu cedness b y semi-red u cedness. In all cases the pro ofs in [8] remain literally the s ame. Due to part (3) of the theorem ab o v e the follo win g n otion will b e imp ortan t to u s. Deﬁnition 3.7 A p olynomia l g ∈ A [ z ; σ ] is called b asic if there exists a unit u ∈ A [ z ; σ ] × suc h that g = P l ∈ T g u ( l ) . According to p art (3) and (4) of Th eorem 3.5, see also Remark 3.6 , a semi-reduced p oly- nomial g is basic if an d only if the matrix G in (3.11) is basic. Let us now h a ve a closer lo ok at semi-redu ced p o lyn omials. F rom (3.10) we see immediately that g is semi-redu ced ⇐ ⇒ ( the num b ers ( a + deg g ( a ) − 1 mo d n ) + 1 , a ∈ T g , are p airwise diﬀerent. (3.12) Moreo v er, one has g ∈ A [ z ; σ ] × and g semi-reduced = ⇒ g ∈ A × . (3.13) Indeed, su pp ose 1 = hg = ( he 1 + . . . + he n ) g = P n l =1 he l g . Since h e l i ∼ = F w e get deg( he l g ) = deg( he l ) + deg( e l g ) and th e leading co eﬃcien t of he l g is in the same ideal as the leading co eﬃcien t of g ( l ) . No w s emi-red u cedness of g shows that n o cancelation of th e leading co eﬃcien ts in P n l =1 he l g is p ossible and thus hg = 1 implies that d eg ( g ( l ) ) = 0 for all l = 1 , . . . , n . It is easy to translate these notions in to the setting of the matrix ring M . Ind eed, deﬁning the s tand ard basis matrices E ab ∈ F n × n via  E ab  ij =  1 if ( i, j ) = ( a, b ) 0 if ( i, j ) 6 = ( a, b )  for a, b = 1 , . . . , n (3.14) w e obtain ξ ( e a ) = E aa . (3.15) Th u s, ξ ( g ( a ) ) = ξ ( e a ) ξ ( g ) = E aa ξ ( g ) (3.16) is a matrix where at most the a -th ro w is nonzero. It is obtained from ξ ( g ) by deleting all other r o ws. This giv es rise to the follo wing d eﬁnition. 9 Deﬁnition 3.8 Let M = ( m ab ) ∈ M . Then w e d eﬁ ne the supp ort of M to b e Supp ( M ) := { a | the a -th ro w of M is non-zero } . W e sa y that M is delay-fr e e if m aa (0) 6 = 0 f or all a ∈ Supp ( M ). By d eﬁ nition Sup p  ξ ( g )  = T g for all g ∈ A [ z ; σ ]. F urthermore, th e very deﬁnition of ξ sho ws that g is dela y-fr ee if and only if ξ ( g ) is. Finall y , we ha ve the implication g is d ela y-free = ⇒ rank F [ t ] ξ ( g ) = rank F ξ ( g ) | t =0 = | T g | . (3.17) This f ollo ws by observing that the n onzero r o ws of ξ ( g ) | t =0 form a matrix in ro w echelo n form with pivot p ositions in th e columns with in d ices in T g . In order to exp r ess semi-reducedn ess in terms of the matrix ξ ( g ) we need the follo wing concept. Deﬁnition 3.9 Let M =  m ab  ∈ M and put d ab := deg( m ab ) for a, b = 1 , . . . , n (where, as u sual, the zero p olynomial h as degree −∞ ). The de gr e e matrix of M is deﬁned as D ( M ) =        nd 11 nd 12 + 1 nd 13 + 2 · · · nd 1 n + n − 1 nd 21 − 1 nd 22 nd 23 + 1 · · · nd 2 n + n − 2 nd 31 − 2 nd 32 − 1 nd 33 · · · nd 3 n + n − 3 . . . . . . . . . . . . . . . nd n 1 − n + 1 nd n 2 − n + 2 nd n 3 − n + 3 · · · nd nn        . Hence D ( M ) ab = nd ab − a + b ∈ N 0 ∪ {−∞} for a, b = 1 , . . . , n . W e call a ro w of D ( M ) trivial i f all en tries are −∞ . Th e matrix M is said to b e semi-r e duc e d if the maxima in the n on-trivial rows of D ( M ) app ea r in d iﬀeren t columns. Ob v ious ly , the trivial rows of D ( M ) corresp o n d to the zero r o ws of M . F ur thermore, w e ha ve the f ollo wing pr op erties. Prop osition 3.10 Let g ∈ A [ z ; σ ] and M = ξ ( g ) = ( m ab ) . F or a, b = 1 , . . . , n put d ab = d eg m ab . (1) In eac h ro w and column of D ( M ) the en tries diﬀeren t from −∞ are pairwise d iﬀerent. In p articular, eac h non-trivial ro w has a u nique m axim um. (2) F or all a, b = 1 , . . . , n we ha ve ξ  g ( a ) e b  = E aa M E bb = m ab E ab and deg  g ( a ) e b  = D ( M ) ab . F urthermore, deg g ( a ) = m ax 1 ≤ b ≤ n D ( M ) ab for all a = 1 , . . . , n (3.18) and deg g = m ax 1 ≤ a ≤ n (deg g ( a ) ) = max 1 ≤ a,b ≤ n D ( M ) ab . (3) g is semi-reduced if and only if M is semi-redu ced. 10 (4) F or a ∈ Su pp ( M ) let δ a := max { d ab | 1 ≤ b ≤ n } and put b a := max { b | d ab = δ a } ; that is, m a,b a is th e r igh tmost entry in the a -th row of M ha ving maximal degree δ a . Then max 1 ≤ b ≤ n D ( M ) ab = D ( M ) a,b a . As a consequence, M is semi-reduced if and only if the indices b a , a ∈ Supp ( M ) , are pairwise diﬀerent . Pr o of: P art (1) is trivial. The ﬁrst assertion of (2) follo ws from (3.15) and the m ulti- plicativit y of ξ . As for th e degree of g ( a ) e b let g b e as in (3.6). Using (3.4) we obtain e a g = X l ≥ 0 n − 1 X i =0 z nl + i c l i, ( i + a − 1 mod n )+1 e ( i + a − 1 mod n )+1 . Notice that for any b = 1 , . . . , n we hav e e ( i + a − 1 mod n )+1 e b = e b ⇐ ⇒ ( i + a − 1 mo d n ) + 1 = b ⇐ ⇒ i = b − a + sgn( a − b ) n while e ( i + a − 1 mod n )+1 e b = 0 for all other v alues of i ∈ { 0 , . . . , n − 1 } . T hus g ( a ) e b = e a g e b = X l ≥ 0 z n  l +sgn( a − b )  + b − a c l b − a +sgn( a − b ) n,b e b . As a consequence, deg( g ( a ) e b ) =  max { l | c l b − a +sgn( a − b ) n,b 6 = 0 } + s gn ( a − b )  n + b − a = n deg( m ab ) + b − a, where the last ident ity follo ws from (3.7). Th is shows deg( g ( a ) e b ) = D ( M ) ab . The last t wo statement s of (2) are a direct consequence. F or p arts (3) and (4) let δ a and b a b e as in (4). Then max {D ( M ) ab | 1 ≤ b ≤ n } = max { nd ab − a + b | 1 ≤ b ≤ n } = nδ a − a + b a = D ( M ) a,b a . No w the rest of (4) is a consequ en ce of Deﬁnition 3.9, whereas (3 ) follo ws from (3.12) along with p art (2) since a + deg g ( a ) − 1 mo d n = b a − 1.  Let u s consider some examples. Example 3.11 Let α b e a pr imitiv e element of F and put n = q − 1. Th en x n − 1 = Q n − 1 i =0 ( x − α i ). Consider the isomorphism φ : A ′ − → A, f 7− → [ f (1) , f ( α ) , . . . , f ( α n − 1 )] of A ′ := F [ x ] / h x n − 1 i with A . Th en the primitive idemp ote nts of A ′ are give n b y e a := γ a Q i 6 = a − 1 ( x − α i ) for a = 1 , . . . , n and some constan ts γ a ∈ F × . Cho ose no w the automorphism σ ∈ Aut F ( A ′ ) d eﬁned via σ ( x ) = α − 1 x . Then one easily c hecks that (3.4) is satisﬁed. This example h as also b een stud ied in [5, Prop. 4.2] (since in that p ap er the automorphism is given by x 7→ αx one has to rep lace α by α − 1 in the idemp ote nts in order to get bac k the ordering as in [5]). In [5, Pr op. 4.2, T h m. 2.1] it has b e en shown that for an y 1 ≤ δ ≤ n − 1 the σ -c y clic su bmo d u le C = p − 1 ( • h g i ), w here g = e 1 P δ i =0 z i , giv es rise to a 1-dimens ional MDS con vo lu tional co d e. Hence the distance of that co de is n ( δ + 1), whic h is the maxim um v alue among all 1-dimensional co des of length n and degree δ (for MDS con volutio n al co d es see [22]). Let us compute an enco der matrix G ∈ F [ z ] 1 × n of C . F r om Theorem 3.5(4) we kno w that G = p − 1 ( g ) is su c h an enco der since g = g (1) . The 11 ﬁrst primitiv e id emp oten t in A ′ is giv en by e 1 = 1 n P n − 1 i =0 x i as one can easily v erify via the isomorp hism φ . Th us, g = 1 n n − 1 X i =0 x i δ X j =0 z j = 1 n δ X j =0 z j n − 1 X i =0 σ j ( x i ) = 1 n δ X j =0 z j n − 1 X i =0 α − j i x i . Using th e map p fr om (2.4) w e obtain G = 1 n  δ X j =0 z j , δ X j =0 z j α − j , . . . , δ X j =0 z j α − ( n − 1) j  = 1 n δ X j =0 z j (1 , α − j , α − 2 j , . . . , α − ( n − 1) j ) . In this s p eciﬁc example the matrix M := ξ ( g ) has a p articularly simple form . In deed, since g = g (1) = P δ i =0 z i e 1+ i and δ ≤ n − 1 we obtain δ +1 z }| { M := ξ ( g ) =   1 1 .. . 1 0 ... 0 0   and D ( M ) =   0 1 ... δ −∞ ... −∞ −∞   . The degree equ ation (3.18) is obvio u s. Notice that the matrix M is idemp ote nt. Th us , g is an idemp ot ent generator of th e left ideal • h g i . Example 3.12 Let q = 5 and n = 4. Consider th e matrix M =     2 + t 1 4 4 0 1 3 0 4 t 2 t 1 + t 1 0 0 0 0     ∈ M . Ob v ious ly , M is d ela y-free. The corresp ond ing p ol yn omial g = ξ − 1 ( M ) ∈ A [ z ; σ ] is give n b y g = 2 e 1 + e 2 + e 3 + z ( e 2 + 3 e 3 + e 4 ) + z 2 (4 e 1 + 4 e 3 ) + z 3 (2 e 2 + 4 e 4 ) + z 4 ( e 1 + e 3 ) , and its comp o n en ts are g (1) = 2 e 1 + ze 2 + 4 z 2 e 3 + 4 z 3 e 4 + z 4 e 1 , g (2) = e 2 + 3 z e 3 , g (3) = e 3 + ze 4 + 4 z 2 e 1 + 2 z 3 e 2 + z 4 e 3 . The matrix, and thus g , are n ot semi-reduced as w e can s ee directly from Prop osi- tion 3.10 (4) or from the d egree matrix D ( M ) =     4 1 2 3 −∞ 0 1 −∞ 2 3 4 1 −∞ −∞ −∞ −∞     . 12 Using t wo steps of elemen tary row op er ations one can br ing M in to semi-reduced form without changing the left ideal • h M i . Ind eed, one easily chec ks that     1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1         1 0 0 0 0 1 0 0 0 3 t 1 0 0 0 0 1         2 + t 1 4 4 0 1 3 0 4 t 2 t 1 + t 1 0 0 0 0     =     2 1 0 0 0 1 3 0 4 t 0 1 1 0 0 0 0     =: ¯ M . (3.19) Since the t wo leftmost matrices are in M × , see Prop osit ion 3.3, w e ha ve • h M i = • h ¯ M i . F u rthermore, ¯ M is semi-redu ced as one can easily see with the help of Pr op ositi on 3.10(4). The corresp onding p ol yn omial ¯ g := ξ − 1 ( ¯ M ) is giv en by ¯ g = 2 e 1 + e 2 + e 3 + z ( e 2 + 3 e 3 + e 4 ) + z 2 4 e 1 , wh ic h is semi-reduced according to Prop o s ition 3.10(3). W e will come bac k to this example later on in Example 4.4. The pro cedu re of semi-reducing matrices in M as in the previous example alwa ys wo r ks. This is shown in the follo wing r esult. Theorem 3.13 A matrix in GL n ( F [ t ]) is said to b e an elemen tary un it of M if it is of an y of the follo wing types: (i) P i 6 = a E ii + αE aa for some α ∈ F × and a = 1 , . . . , n , (ii) I n + t N αE ab for some N ≥ 0 , α ∈ F and 1 ≤ a < b ≤ n , (iii) I n + t N αE ab for some N > 0 , α ∈ F and 1 ≤ b < a ≤ n . Then eac h matrix in M can b e br ough t in to semi-redu ced form via left multiplicatio n b y ﬁnitely many elemen tary un its of M . As a consequence, eac h unit M ∈ M × can b e written as a pro d uct of elemen tary un its of M . Pr o of: It is trivial that eac h of the matrices in (i) – (iii) is ind eed in M × , and that th e in verse of suc h an elemen tary u nit is an elemen tary unit of the same t y p e. Let n o w M = ( m ab ) ∈ M a n d a s sume th at M is not semi-reduced. Then, according to Prop osition 3.10 th ere exist indices ( a, c ) and ( b, c ) where b > a suc h that D ( M ) a,c and D ( M ) b,c are the maxima in th e a -th and b -th r o w of D ( M ), resp ectiv ely . Using th e deﬁnition of the matrix D ( M ) one easily chec ks that for b > a and j = 1 , . . . , n D ( M ) bj < D ( M ) aj ⇐ ⇒ deg( m bj ) ≤ deg ( m aj ) . (3.20) W e wan t to transform M into semi-reduced form via left multi p lication by elemen tary units of M . Consid er ﬁ rst the case D ( M ) bc ≤ D ( M ) ac . Pu t ˆ M = ( ˆ m ij ) = ( I n + t N αE ab ) M wh ere N := deg( m ac ) − deg( m bc ) ≥ 0 an d α ∈ F is such that deg ( ˆ m ac ) = deg ( m ac + t N αm bc ) < deg( m ac ). This is p ossib le due to (3.20). Then D ( ˆ M ) ac < D ( M ) ac . F u rthermore, for j 6 = c w e hav e ˆ m aj = m aj + t N αm bj and thus D ( ˆ M ) aj = n deg( ˆ m aj ) − a + j ≤ n max { deg( m aj ) , N + deg ( m bj ) } − a + j = max { n deg( m aj ) − a + j, nN + n deg( m bj ) − a + j } = max {D ( M ) aj , D ( M ) bj + nN + b − a } < max {D ( M ) ac , D ( M ) bc + nN + b − a } , 13 where the last inequalit y h olds tr ue due to the uniqueness of the row maxima in D ( M ) (see Prop osit ion 3.10(1)). Since D ( M ) bc + n N + b − a = n deg( m bc ) − b + c + nN + b − a = n deg( m ac ) − a + c = D ( M ) ac the ab o ve results in D ( ˆ M ) aj < D ( M ) ac for all j = 1 , . . . , n . If D ( M ) bc > D ( M ) ac , then put ˆ M := ( I n + t N αE ba ) M where N > 0 and α ∈ F are c h osen suc h that deg( m bc + t N αm ac ) < d eg( m bc ). Again, this is p ossible by virtue of (3.20). Argu- ing the same wa y we obtain lik ewise D ( ˆ M ) bj < D ( M ) bc for all j = 1 , . . . , n . S u mmarizing w e see that if M is not semi-reduced we m a y app ly one of the tw o transform ations give n ab o ve, and they b o th s tr ictly decrease the maxim u m v alue in one of the ro ws of D ( M ) while all other r o ws remain unc hanged. Altogether this results in a red u ction p r o cedure that must stop after ﬁnitely many steps with a semi-reduced matrix. The last statemen t of the th eorem follo ws directly from (3.13) together with the use of elemen tary un its of Typ e (i).  One easily chec ks that the element ary un its in Theorem 3.13 corresp o n d to the un its ξ − 1  X i 6 = a E ii + αE aa  = αe a + X i 6 = a e i , ξ − 1  I n + t N αE ab  = 1 + z N n + b − a αe b , in A [ z ; σ ] . Of course, in the seco n d case w e ha ve a 6 = b and N > 0 if b < a . These units in A [ z ; σ ] ha ve b een studied in detail in the pap er [6]. In a more general con text it has b een sho wn in [6, Lem. 3.7] that 1 + z d αe b , wh ere d > 0, is a unit if and only if n ∤ d , and that eac h u nit in A [ z ; σ ] can b e written as the pro d u ct of ﬁnitely man y un its of the typ es ab ov e. This corresp onds exactly to our last statement in Theorem 3.13 ab o ve. With that theorem w e ev en see that left reduction to a semi-redu ced p olynomial in th e sk ew-p o lyn omial r ing A [ z ; σ ], cf. [8, Cor. 4.13], simp ly means elemen tary ro w redu ction of the m atrices in M . The follo wing r esult tells us as to when a matrix in M can b e completed to a un it in M b y ﬁlling in suitable entrie s in the zero r o ws. This result w ill b e imp ortant later on w hen studying w hether a cyclic s ubmo d ule is a direct summand . Lemma 3.14 Let M ∈ M b e dela y-free and S upp ( M ) = { i 1 , . . . , i k } , where i 1 < . . . < i k . Denote the r ows of M b y M 1 , . . . , M n . T hen the follo wing are equiv alen t. (a) Th e matrix f M := M i 1 . . . M i k ! is b asic. (b) Th ere exist v ectors c M i ∈ F [ t ] 1 × n for i ∈ { 1 , . . . , n }\{ i 1 , . . . , i k } su c h that th e m atrix N =    N 1 . . . N n    ∈ F [ t ] n × n , where N i := ( M i if i ∈ { i 1 , . . . , i k } c M i if i ∈ { 1 , . . . , n }\{ i 1 , . . . , i k } is in M × . Since ev ery matrix in M × is d ela y-free it is clear that the dela y-fr eeness is n ecessary for the imp lication (a) = ⇒ (b) to b e tru e. 14 F or the pro of the follo wing suggestiv e notation will b e helpful. Arb itrary en tries of a matrix in F [ t ] r × n will b e ind icated by an asterisk ∗ , whereas en tr ies that are multiples of t will b e denoted by the symb ol h t i . Thus, the element s of M are jus t the matrices of the form    ∗ ∗ ··· ∗ h t i ∗ ∗ . . . . . . . . . . . . h t i ··· h t i ∗    . Pr o of: Only the implication “(a) ⇒ (b)” requires pro of. It is a w ell-kno wn fact that, since the k -minors of f M are coprime, there exists a matrix c M ∈ F [ t ] ( n − k ) × n suc h that N :=  f M c M  ∈ GL n ( F [ t ]). T h erefore, N is of the form        h t i ··· m i 1 i 1 ∗ ··· ∗ h t i ··· ··· ··· m i 2 i 2 ∗ ··· ∗ . . . . . . h t i ··· ··· ··· · ·· ··· ·· · m i k i k ∗ ··· ∗ c M        , where all en tries in the upp er blo c k and to the left of m i j ,i j are of th e t yp e h t i . W e will show no w that w ith suitable r o w op e r ations w e can transform N in to a matrix b eing in M without altering the ro w s of the upp er part f M . By d ela y-freeness of M w e h av e m i j i j (0) 6 = 0 for all j = 1 , . . . , k . Hence, by adding suitable multiples of rows of f M to c M w e can transform N in to a matrix of the form            h t i ··· m i 1 i 1 ∗ ··· ∗ h t i ··· ··· ··· m i 2 i 2 ∗ ··· ∗ . . . . . . h t i ··· ··· ··· ··· ··· ·· · m i k i k ∗ ··· ∗ h t i h t i h t i ∗ . . . ∗ . . . ∗ . . . ∗ h t i h t i h t i            ∈ GL n ( F [ t ]) , (3.21) where th e upp er part f M is unaltered. Th erefore, w e ma y assume without loss of generalit y that N is as in (3.21). F or the n ext step observe that for an y column v ector a ∈ F [ t ] l × 1 there exists U ∈ GL l ( F [ t ]) su c h that U a is of the form    ∗ h t i . . . h t i    . Th u s, w orking consecutiv ely from the left to the righ t and ap p lying suitable elemen tary ro w op e r ations on the low er blo ck of N w e can br ing c M into the form 15 c M =                     ∗ h t i h t i h t i . . . . . . . . . . . . ∗ h t i . . . h t i · · · h t i h t i ∗ h t i . . . . . . . . . h t i . . . . . . . . . . . . . . . . . . ∗ h t i . . . . . . . . . . . . h t i h t i . . . . . . . . . . . . . . . . . . . . . . . . h t i · · · h t i h t i h t i h t i h t i                            i 1 − 1 rows        i 2 − i 1 − 1 rows | {z } i 1 columns | {z } i 2 − i 1 columns while f M do es not c hange. Assume that N =  f M c M  is of this form and recall that th e j -t h row of f M is giv en b y M i j , j = 1 , . . . , k . No w, moving for j = 1 , . . . , k the j -th ro w of f M to the b ottom of the j -th blo c k of c M we can f orm a matrix N ′ where the i j -th row is giv en by M i j for j = 1 , . . . , k and w h ere we kee p the ord ering of all remaining ro ws of N . This wa y , the entries in c M that are exp licitly indicated by asterisks and the en tr ies m i j i j , j = 1 , . . . , k , will app e ar on th e d iagonal of N ′ , while all en tries b elo w the diagonal will b e in h t i . Hence N ′ ∈ GL n ( F [ t ]) ∩ M and Prop ositi on 3.3 completes th e p ro of.  Remark 3.15 Essent ially all of the r esults of this section are true withou t the requiremen t of n dividing q − 1. Ind eed, ju st consider A an d σ as in (3.1) and (3.3). Section 4 of [8] remains true in this case s ince it wa s sole ly based on A b e in g a direct pr o duct of ﬁelds. The only part that n eeds extra pro o f is part (3) of T heorem 3.5, and that can b e accomplished using rin g theoretic metho ds. Part (4) of that theorem do es not ha ve a meaning in this more general setting since A is not isomorphic to F [ x ] / h x n − 1 i anymore, and th us the map p do es not exist. W e will b rieﬂy come bac k to this situation in Section 6 . 4 Construction and Existence of σ -CCC’s In this section w e will apply th e results obtained so far in order to construct σ -CCC’s, and we will discuss some existence issues. As b e f ore, let A and σ b e as in (3.1) and (3.3) and let n | ( q − 1). Moreo ver, put M basic :=  M ∈ M     M dela y-fr ee and the nonzero ro w s of M form a b asic matrix  . (4.1) Let u s summarize the previous results in the follo wing form. W e also think it is w orthwhile p ointi n g out th at the prop erty of a mo d ule b e in g a direct sum mand as an F [ z ]-su bmo du le is equiv alent to b eing a direct summand as a left ideal in the sk ew p o lyn omial r in g. Recall the m ap p f r om (2.4). 16 Theorem 4.1 Let g ∈ A [ z ; σ ] b e dela y-fr ee and semi-reduced. Then th e follo wing are equiv alent. (i) • h g i is a direct summand of the left F [ z ] -modu le A [ z ; σ ] (th us, p − 1 ( • h g i ) is a σ -CCC). (ii) • h g i is a d irect su mmand of the ring A [ z ; σ ] . (iii) ξ ( g ) ∈ M basic . Pr o of: The equiv alence of (i) and (ii) h as b een pro ved in [6, Rem. 2.10], whereas (i) ⇐ ⇒ (iii) follo ws from Lemma 3. 14, Theorem 3.5(3), an d the f act that ξ is an iso- morphism.  Notice th at (iii) giv es us an easy wa y of chec k in g whether a giv en left ideal is a dir ect summand (th us a σ -CC C ) since basicness of a matrix in F [ t ] k × n can, for instance, b e c hec ked b y testing whether its k -minors are coprime. As a consequence, σ -CCC ’s are in one-one corresp o n dence w ith the left ideals • h M i ⊆ M where M ∈ M basic is semi-redu ced. Prop o s ition 3.10(3) and Theorem 3.5(4), see also Remark 3.6, tell us immediately th e algebraic parameters of th e co de, that is, the dimension, the F orney indices and degree. This is s ummarized in the next result. Corollary 4.2 Let M ∈ M basic b e semi-reduced and let Su pp ( M ) = { i 1 , . . . , i k } , wh ere i 1 < . . . < i k . Then C := p − 1 ( • h ξ − 1 ( M ) i ) ⊆ F [ z ] n is a k - d imensional σ -CC C with F orney indices giv en by ν l := max 1 ≤ j ≤ n D ( M ) i l ,j for l = 1 , . . . , k . Example 4.3 L et us consider again the s etting of Example 3.11 where n = q − 1 and σ ( x ) = α − 1 x with some primitiv e elemen t α of F . I n that examp le w e presente d some 1- dimensional σ -cyclic MDS con v olutional co d es. These co des can b e generalized as follo ws. Let g = ( e 1 + e 2 )(1 + z ). Then M = ξ ( g ) =   1 1 0 0 ... 0 0 1 1 0 ... 0 0   , D ( M ) =   0 1 −∞ −∞ ... −∞ 0 0 1 −∞ ... −∞ −∞   . Ob v ious ly , M ∈ M basic and h ence the previous corollary guaran tees that C := p − 1 ( • h g i ) ⊆ F [ z ] n is a σ -CCC. F urthermore, M , and thus g , is semi-reduced. As a consequence, the co de C is a 2-dimensional co de in F [ z ] n with b oth F orney ind ices equal to 1 and d egree 2. In other wo r ds, C is a unit m emory co de. W e w ill sh ow no w that these co des are optimal in the sense that they ha ve the largest distance among all 2-dimens ional co des with F orney indices 1 , 1 and length n = q − 1 o ver F q . According to [9, Prop. 4. 1] (see also [6 , Eq. (1.3)]) the largest p o ss ib le distance for co des with these parameters, called the Griesmer b oun d , is giv en by the num b e r 2( n − 1). In order to compute the actual distance of C we n eed an enco der matrix. S ince the su p p ort of g is T g = { 1 , 2 } and g (1) = e 1 + z e 2 , g (2) = e 2 + z e 3 , Theorem 3.5(4) tells us that a m inimal enco der is giv en b y th e matrix G = G 0 + G 1 z where G 0 =  p − 1 ( e 1 ) p − 1 ( e 2 )  , G 1 =  p − 1 ( e 2 ) p − 1 ( e 3 )  ∈ F 2 × n . F u rthermore, G is basic and minimal, and th us r ank G 0 = rank G 1 = 2. No w it is easy to see that th e t wo blo ck co des generated by G 0 and G 1 , resp ectiv ely , are MDS cod es, 17 that is, they b oth ha v e distance n − 1. Ind eed, recall from Example 3.11 that e a = γ a Q i 6 = a − 1 ( x − α i ) for a = 1 , . . . , n . Thus, in the ring A ′ = F [ x ] / h x n − 1 i the ideal h e 1 , e 2 i is identic al to h f i , where f = Q n − 1 i =2 ( x − α i ). As a consequence, the cyclic block co d e im G 0 = p − 1 ( h e 1 , e 2 i ) = p − 1 ( h f i ) ⊆ F n has d esigned distance n − 1. The second b lo c k co d e h e 2 , e 3 i is s imply the image of the ﬁrst one und er the map σ . S ince σ is we ight-preserving this sho w s that the second code h as distance n − 1, to o. But no w it is clear that the con vo lutional co de C has d istance 2( n − 1) since for eac h message u = P N j =0 u j z j ∈ F [ z ] 2 , where u 0 6 = 0 6 = u N , the corresp o n ding cod ew ord uG has constan t term u 0 G 0 and highest co eﬃcien t u N G 1 , b o th of weigh t at least n − 1. In the same w a y one can pro ceed and consider the unit memory code generated by the p olynomial g = ( e 1 + e 2 + e 3 )(1 + z ). Again, the matrix M := ξ ( g ) sho w s that g is semi-reduced and b asic and thus C = p − 1 ( • h g i ) is a 3-dimens ional σ -CCC with all F orn ey indices b eing 1. In this case, [9 , Pr op. 4.1] tells us that, if n ≥ 6, the Griesmer b ound for these p arameters is give n by the num b er 2( n − 2) + 1. In th e same w ay as ab o ve one can sho w that the codes ju st constructed hav e distance 2( n − 2), that is, they fail the Griesmer b ound by 1. Pro ceeding in the s ame w ay for arb itrary k ≤ n 2 , one obtains k -d im en sional unit memory σ -CCC’s h aving d istance 2( n − k + 1) whic h is k − 2 b e low th e corresp onding Griesmer b oun d . Example 4.4 L et q = 5 and n = 4 and ¯ g b e as in Example 3.12. W rite F = F 5 . One easily c hec ks that the matrix ¯ M = ξ (¯ g ) giv en in (3.19) is in M basic . Thus, the sub mo dule C = p − 1 ( • h ¯ g i ) ⊆ F [ z ] 4 is a 3-dimensional σ -CC C with F orney indices 1 , 1 , 2. In ord er to compute a minimal enco der G ∈ F [ z ] 3 × 4 of C w e will apply Th eorem 3.5(4). F rom ¯ g as giv en in Example 3.12 we obtain the comp onen ts ¯ g (1) = 2 e 1 + z e 2 , ¯ g (2) = e 2 + 3 z e 3 , ¯ g (3) = e 3 + z e 4 + 4 z 2 e 1 . I den tifyin g f ∈ F [ x ] / h x 4 − 1 i with [ f (1 ) , f (2) , f (4) , f (3)] ∈ A and using the m ap p f r om (2.4) we arriv e at the m inimal enco d er matrix G =   p − 1 ( ¯ g (1) ) p − 1 ( ¯ g (2) ) p − 1 ( ¯ g (3) )   =   4 z + 3 2 z + 3 z + 3 3 z + 3 2 z + 4 3 z + 2 2 z + 1 3 z + 3 z 2 + 4 z + 4 z 2 + 3 z + 1 z 2 + z + 4 z 2 + 2 z + 1   . Using some compu ter algebra routine one c hec ks that this co de attains the Griesmer b ound (see [6, Eq. (1.3)]) . Pr ecisely , its d istance is 6, whic h is the largest distance p ossible for an y 3-dimensional co de of length 4 o ver F 5 with the same F orn ey in dices. Let us n ow turn to the existence of σ -CCC ’s with prescrib ed algebraic parameters. Corol- lary 4.2 raises the question w h ether for all 1 ≤ k ≤ n − 1 and ν 1 , . . . , ν k ∈ N 0 there exists a k -d imensional σ -CCC in F [ z ] n with F orney indices ν 1 , . . . , ν k . The rest of this section will b e devot ed to this problem. W e will s tart with sho wing that this problem can b e split in to t wo su bproblems one of whic h is purely com binatorial. Put ˆ D :=        0 1 2 . . . n − 2 n − 1 n − 1 0 1 . . . n − 3 n − 2 . . . . . . . . . . . . . . . . . . 2 3 4 . . . 0 1 1 2 3 . . . n − 1 0        , (4.2) 18 th us , ˆ D ab = ( b − a, if b ≥ a n + b − a, if b < a. ) = b − a mo d n. (4.3) The role of the matrix ˆ D is explained by the fact that for eve r y matrix M = ( m ab ) ∈ M w e hav e D ( M ) ab = ( n d eg m ab + ˆ D ab if b ≥ a, n (deg m ab − 1) + ˆ D ab if b < a. (4.4) The com binatorial p roblem w e need to consider sh o ws some close resemblance w ith th e classical ro ok p roblem. Problem 4.5 (Mo diﬁed Ro ok Problem) Let r 1 , . . . , r k b e an y (not necessarily diﬀer- en t) num b e r s in { 0 , . . . , n − 1 } . Can w e ﬁ n d these num b ers in the matrix ˆ D su c h that they app ear in pairwise diﬀerent ro ws and pairwise d iﬀeren t columns? In other words, can we ﬁnd distin ct n u m b ers i 1 , . . . , i k and distin ct n u m b ers j 1 , . . . , j k , all in the set { 1 , . . . , n } , suc h that ˆ D i l ,j l = r l for l = 1 , . . . , k . (4.5) The follo w in g sligh t reformulation will come h andy f or our pur p oses. Remark 4.6 If Problem 4.5 is solv able for r 1 , . . . , r k then we can ﬁnd these num b ers ev en in the ﬁ rst n − 1 ro ws of the m atrix ˆ D . In other w ords, (4.5) is tr u e for some distinct n u m b ers i 1 , . . . , i k ∈ { 1 , . . . , n − 1 } and distinct num b ers j 1 , . . . , j k ∈ { 1 , . . . , n } . Indeed, supp ose we ha ve a solution to 4. 5 , that is, ˆ D i l ,j l = r l for l = 1 , . . . , k . Then we ma y construct a second solution as follo ws. T here exists s ome α ∈ { 1 , . . . , n } such that i l 6 = α for all l . Put a l = ( i l − α − 1 mo d n ) + 1 and b l = ( j l − α − 1 mo d n ) + 1. S ince the n u m b ers ( i 1 mo d n ) , . . . , ( i k mo d n ) are pairwise diﬀerent the same is true for a 1 , . . . , a k . Lik ewise b 1 , . . . , b k are pairwise diﬀeren t. Of course, a l , b l ∈ { 1 , . . . , n } for all l = 1 , . . . , k . Moreo v er, b y construction a l 6 = n for all l = 1 , . . . , k , and up on using (4.3) w e obtain ˆ D a l ,b l = b l − a l mo d n = j l − i l mo d n = ˆ D i l ,j l = r l for l = 1 , . . . , k . In the next section w e w ill study Problem 4.5 in some more detail. Even though w e are not able to pro vid e a p ro of of the solv abilit y for general num b ers r 1 , . . . , r k w e will consider some s p ecial cases w here we present a complete p ro of. W e would lik e to exp r ess our strong b elief that the problem can b e solve d for all give n data r 1 , . . . , r n − 1 . This has b een unders cored b y a routine c hec k with Maple conﬁrming our conjecture for all n ≤ 10. The second problem we need to consider has an aﬃrmativ e answe r an d th us can b e stated as a theorem. The pro of will b e present ed at the end of this s ection. Theorem 4.7 Let j 1 , . . . , j n − 1 ∈ { 1 , . . . , n } b e pairwise diﬀeren t and d 1 , . . . , d n − 1 ∈ N 0 b e s uc h that j i < i = ⇒ d i > 0 . (4.6) Then there exists a basic matrix M = ( m ij ) ∈ F [ t ] ( n − 1) × n with the follo wing p rop erties: (i) deg m ij ≤ d i for j < j i , 19 (ii) deg m ij = d i for j = j i , (iii) deg m ij < d i for j > j i , (iv) m ii (0) = 1 for all i , (v) m ij (0) = 0 for j < i . Notice th at the prop e r ties (i) – (iii) simp ly tell us that the i -th ro w degree is giv en b y d i and the rightmo s t entry with degree d i is in column j i . Moreo ver, observ e that w ithout (4.6) the requiremen ts (ii) and (v) would not b e compatible. Using Prop osition 3.10 we see that if we extend M by a zero row at the b ott om we obtain a semi-reduced matrix in M basic . No w we can sh o w the follo wing. Theorem 4.8 Let 1 ≤ k ≤ n − 1 and ν 1 , . . . , ν k ∈ N 0 . P ut r l = ν l mo d n for l = 1 , . . . , k . If Problem 4.5 is solv able for r 1 , . . . , r k then there exists a k -dimensional σ -CC C in F [ z ] n with F orney indices ν 1 , . . . , ν k . As a consequence, if Problem 4.5 is solv able for all r 1 , . . . , r n − 1 ∈ { 0 , . . . , n − 1 } then for all 1 ≤ k ≤ n − 1 and all ν 1 , . . . , ν k ∈ N 0 there exists a k -dimensional σ -CCC in F [ z ] n with F orn ey in dices ν 1 , . . . , ν k . Pr o of: Let k ∈ { 1 , . . . , n − 1 } and ν 1 , . . . , ν k ∈ N 0 . W r ite ν l = ˆ d l n + r l , wh ere ˆ d l ∈ N 0 and 0 ≤ r l ≤ n − 1. By assum p tion and Remark 4.6 th er e exist distinct i 1 , . . . , i k ∈ { 1 , . . . , n − 1 } an d distinct j 1 , . . . , j k ∈ { 1 , . . . , n } su ch that ˆ D i l ,j l = r l for l = 1 , . . . , k . Pic k 1 ≤ i k +1 , . . . , i n − 1 ≤ n − 1 and 1 ≤ j k +1 , . . . , j n − 1 ≤ n su ch that i 1 , . . . , i n − 1 as we ll as j 1 , . . . , j n − 1 are pairwise d iﬀeren t. Deﬁne r l := ˆ D i l ,j l for l = k + 1 , . . . , n − 1 and put ˆ d l := 0 and ν l := r l for l = k + 1 , . . . , n − 1. No w w e re-index the num b ers r 1 , . . . , r n − 1 in order to obtain ˆ D l,j l = r l for l = 1 , . . . , n − 1 . (4.7) Deﬁne d l :=  ˆ d l if j l ≥ l, ˆ d l + 1 if j l < l. (4.8) Then (4.6) is true and Theorem 4.7 guaran tees the existence of a matrix ˜ M = ( m ij ) ∈ F [ t ] ( n − 1) × n satisfying (i) – (v). Extending ˜ M by a zero row at the b ottom results in a semi-reduced matrix M ∈ M basic , see also Prop o s ition 3.10(4) . Using (4.4), w e see that the m axima in the ﬁ rst n − 1 ro ws of D ( M ) are giv en by D ( M ) l,j l = ( nd l + ˆ D l,j l = n ˆ d l + r l = ν l , if j l ≥ l, n ( d l − 1) + ˆ D l,j l = n ˆ d l + r l = ν l , if j l < l. Finally , deleting th e ro w s of M corresp ond ing to the n − 1 − k artiﬁcially added in dices ν k +1 , . . . , ν n − 1 (in the original ordering) w e obtai n a semi-redu ced matrix N ∈ M basic , and Corollary 4.2 sho ws that p − 1 ( • h ξ − 1 ( N ) i ) is a k - d imensional co d e with F orney indices ν 1 , . . . , ν k .  Example 4.9 S upp ose we wan t a 3-dimensional σ -CCC C ⊆ F 5 [ z ] 4 with F orney indices 4 , 3 , 3. Th u s, q = 5 and n = 4. Th e r emainders mo du lo n of the desired F orney indices 20 are 0 , 3 , 3, and by insp ection w e ﬁ nd ˆ D 1 , 4 = 3 =: r 1 , ˆ D 2 , 1 = 3 =: r 2 , ˆ D 3 , 3 = 0 =: r 3 . Th u s, (4.7) is true for ( j 1 , j 2 , j 3 ) = (4 , 1 , 3) . Th is giv es us the ordering ν 1 = 3 , ν 2 = 3 , ν 3 = 4 of th e F orn ey indices, and w e ha ve ˆ d 1 = ˆ d 2 = 0 and ˆ d 3 = 1. F ollo wing (4.8) w e put d 1 = 0 , d 2 = 1 , d 3 = 1. Th en w e ha ve all d ata f or Theorem 4.7. One easily sees that the matrix M =   1 0 0 1 t 1 0 0 0 0 1 + t 1   is basic and satisﬁes (i) – (v) of that theorem. Add ing a zero row at the b ottom giv es us a semi-reduced m atrix N ∈ M basic , and using Corollary 4.2 w e see that C = p − 1  • h ξ − 1 ( N ) i  is a 3-dimensional σ -CCC in F 5 [ z ] 4 with F orney in dices 3 , 3 , 4. An enco der matrix of C can b e obtained as follo ws. The r o ws of M along with the isomorphism ξ result in the comp onent p olynomials g (1) = e 1 + z 3 e 4 , g (2) = e 2 + z 3 e 1 , g (3) = e 3 + z e 4 + z 4 e 3 . Us- ing the same identiﬁca tion of A with F 5 [ z ] / h x 4 − 1 i as in Example 4.4 we obtain f rom Theorem 3.5(4) that G =   p − 1 ( g (1) ) p − 1 ( g (2) ) p − 1 ( g (3) )   =   4 z 3 + 4 3 z 3 + 4 z 3 + 4 2 z 3 + 4 4 z 3 + 4 4 z 3 + 2 4 z 3 + 1 4 z 3 + 3 4 z 4 + 4 z + 4 z 4 + 3 z + 1 4 z 4 + z + 4 z 4 + 2 z + 1   is a minimal encod er of C . It sh ould b e p oi nted out that the distance of this cod e is far from b eing optimal. This is d ue to the ab u ndance of zeros in th e matrix M causin g many zero co eﬃcien ts in the matrix G . Let us brieﬂy men tion the follo wing conv erse of Theorem 4.8. Indeed, it is easy to see that the existence of ( n − 1)-dimensional σ -CCC’s with arb itrarily prescrib ed F orney indices implies the solv abilit y of Pr oblem 4.5 for an y giv en num b ers r 1 , . . . , r n − 1 ∈ { 0 , . . . , n − 1 } . In more detail, the existence of su c h co des imp lies the existence of s emi-reduced p olyno- mials g ∈ A [ z ; σ ] with su pp ort satisfying | T g | = n − 1 and arbitrarily giv en degrees of its nonzero comp onen ts. Using Prop osit ion 3.10(3) this sho w s the solv abilit y of Problem 4.5 for k = n − 1, and thus for arbitrary k ∈ { 1 , . . . , n − 1 } . While the general formulation of Theorem 4.8 is based on the assum ption th at we can solv e Prob lem 4.5 we ha ve some sp eciﬁc cases with fully established existence results. Theorem 4.10 Let 1 ≤ k ≤ n +1 2 . Then for all ν 1 , . . . , ν k ∈ N 0 there exists a k - dimensional σ -CCC h a ving F orney in dices ν 1 , . . . , ν k . Pr o of: Using Theorem 4.8 it suﬃces to sh o w that Pr oblem 4.5 can b e solv ed for any giv en n umbers r 1 , . . . , r k ∈ { 0 , . . . , n − 1 } if k ≤ n +1 2 . First of all, it is clear that there exists j 1 suc h that ˆ D 1 ,j 1 = r 1 , and w e will p ro ceed by in duction. T hus, let u s assume that w e found distinct in d ices i 1 , . . . , i k − 1 and j 1 , . . . , j k − 1 suc h that ˆ D i l ,j l = r l for l = 1 , . . . , k − 1. After a suitable p ermutatio n of the rows and columns of ˆ D we obtain a matrix ˜ D =  ˜ D 1 ˜ D 2 ˜ D 3 ˜ D 4  where ˜ D 1 = diag( r 1 , . . . , r k − 1 ) ∈ Z ( k − 1) × ( k − 1) 21 and wh ere th e other matrices are of ﬁtting sizes. In particular, the matrix ˜ D 3 is of size ( n − k + 1) × ( k − 1). Since the en tries of eac h ro w (resp. column) of ˜ D are pairwise diﬀerent and, b y assump tion, n − k + 1 > k − 1 th er e exists at least one r o w of ˜ D 3 that do e s not con tain the en tr y r k . But then r k m us t o c cur in the submatrix ˜ D 4 , and therefore we hav e found r 1 , . . . , r k in the matrix ˆ D in pairwise diﬀerent columns and rows.  One migh t wonder whether the last result can b e extended to co des with arbitrary dimen- sion by u sing dual co d es. Recall that the dual C ⊥ of a k -d im en sional co de C ⊆ F [ z ] n has dimension n − k . Ho w ever, it is a well -kn o wn f act in con vol u tional co din g theory that the F orn ey in dices of the dual co d e are n ot determined b y the F orn ey indices of the giv en co de. This is also true in the s p ecia l case of σ -CCC’s as one can easily see b y some examples. As a consequence, Theorem 4.10 do es not imp ly any existence results for co des with higher dimension. In the next sect ion w e w ill show that we can solv e Problem 4.5 for parameters r 1 , . . . , r n − 1 that attain at most tw o d iﬀeren t v alues, see P r op osition 5.2. Consequently , we h a ve the follo wing resu lt. Theorem 4.11 Let 1 ≤ k ≤ n − 1 and ν 1 , . . . , ν k ∈ N 0 . I f |{ ν 1 mo d n, . . . , ν k mo d n }| ≤ 2 then there exists a k -dimensional σ -C C C in F [ z ] n with F orney indices ν 1 , . . . , ν k . W e will close this section with the Pr o of of Theorem 4.7: W e assu me that F is an y ﬁnite ﬁeld and n ≥ 2. The follo wing n otation will b e helpful. F or a matrix A ∈ F [ t ] n × ( n +1) and l = 1 , . . . , n + 1 let A ( l ) denote the n -minor of A obtained b y omitting column l . Recall that A is basic if and only if the p olynomials A (1) , . . . , A ( n +1) are coprime. W e w ill p ro ve eve n more than stated in Theorem 4.7. W e will sho w that for th e giv en d ata there exists a matrix M satisfying the r equ iremen ts of the theorem and with the follo wing additional pr op erties: (vi) The only nonzero elemen ts b elo w the d iagonal of M are at the p o sitions ( i, j i ) where j i < i and they are of th e form t d i . T his, of course, implies (v). (vii) If j i < i , then the only nonconstant elemen t in the i -th r ow is at p osit ion ( i, j i ). Using (vi), part (vii) tells us that if j i < i then all elemen ts at p o s itions ( i, j ) w here j ≥ i are constant. W e w ill pro ceed by ind uction on n . Let n = 2 and j 1 , d 1 b e giv en. F or j 1 = 1 th e matrices M = (1 + t d 1 , 1), if d 1 > 0, and M = (1 , 0), if d 1 = 0, are basic and hav e the p rop erties (i) – (vii). If j 1 = 2, the matrix M = (1 , t d 1 ) satisﬁes all requirements. Let no w n ≥ 2 and assume that for all p ossible j 1 , . . . , j n − 1 and d 1 , . . . , d n − 1 ∈ N 0 a basic matrix M ∈ F [ t ] ( n − 1) × n satisfying (i) – (vii) exists. Throu gh ou t this pro o f w e will call suc h a matrix a solution for the parameters ( j 1 , . . . , j n − 1 ; d 1 , . . . , d n − 1 ). Notice that due to (iv) and (v) we hav e t ∤ M ( n ) . Assume no w w e ha v e p airwise diﬀerent indices j 1 , . . . , j n ∈ { 1 , . . . , n + 1 } and in tegers d 1 , . . . , d n ∈ N 0 suc h th at (4.6) holds true. W e will show the existence of a b asic matrix M ∈ F [ t ] n × ( n +1) satisfying (i) – (vii) separately f or eac h of the follo win g cases. 22 Case 1: j n = n + 1. Then j 1 , . . . , j n − 1 ≤ n and by induction hyp othesis there exists a solution ˆ M = ( ˆ m 1 , . . . , ˆ m n ) ∈ F [ t ] ( n − 1) × n for the p arameters ( j 1 , . . . , j n − 1 ; d 1 , . . . , d n − 1 ). Put M =  ˆ m 1 · · · ˆ m n − 1 ˆ m n 0 0 · · · 0 1 t d n  ∈ F [ t ] n × ( n +1) . Since M n,n = 1 and ˆ M satisﬁes (i) – (vii) the same is true for M . Moreo v er, M is basic as w e can see by considering th e n -minors. Ind eed, w e hav e M ( i ) = ± t d n ˆ M ( i ) for i = 1 , . . . , n and M ( n +1) = ± ˆ M ( n ) . No w the coprimeness of M (1) , . . . , M ( n +1) follo ws fr om the b asicness of ˆ M along w ith the fact that t ∤ ˆ M ( n ) . Case 2: j n = n and d n = 0. F or i = 1 , . . . , n − 1 pu t ˆ j i =  j i if j i < n n if j i = n + 1 (notice that the case j i = n +1 need not o c cur ). Th en the in dices ( ˆ j 1 , . . . , ˆ j n − 1 ; d 1 , . . . , d n − 1 ) satisfy (4.6) and th u s there exists a solution ˆ M = ( ˆ m 1 , . . . , ˆ m n ) ∈ F [ t ] ( n − 1) × n for these parameters. P ut M =  ˆ m 1 · · · ˆ m n − 1 0 ˆ m n 0 · · · 0 1 0  ∈ F [ t ] n × ( n +1) . It is easy to see that M is a solution for ( j 1 , . . . , j n ; d 1 , . . . , d n ). Case 3: j n = n and d n > 0. Cho ose ˆ M as in the previous case and put M =  ˆ m 1 · · · ˆ m n − 1 ˆ m n ˆ m n 0 · · · 0 1 + t d n 1  ∈ F [ t ] n × ( n +1) . In this case the n -minors of M are given b y M ( n +1) = ± (1 + t d n ) ˆ M ( n ) and M ( n ) = ± ˆ M ( n ) , whereas for i = 1 , . . . , n − 1 w e ha ve by expansion alo n g the last row M ( i ) = ±  (1 + t d n ) ˆ M ( i ) − ˆ M ( i )  = ± t d n ˆ M ( i ) . Again, basicness of ˆ M along with th e fact that t ∤ ˆ M ( n ) implies basicness of M . It is easy to see that M satisﬁes the prop e r ties (i) – (vii). Case 4: j n =: α < n . Then d n > 0. W e h a v e to d istinguish fur ther cases. (a) If j i ≤ n for all i = 1 , . . . , n − 1 let ˆ M = ( ˆ m 1 , . . . , ˆ m n ) b e a solution for th e parameters ( j 1 , . . . , j n − 1 ; d 1 , . . . , d n − 1 ) and put M =  ˆ m 1 · · · ˆ m α · · · ˆ m n 0 t d n 1 1  ∈ F [ t ] n × ( n +1) . It is easy to see that M is b asic and s atisﬁes (i) – (vii). (b) S upp ose no w there exists some index β suc h that j β = n + 1 and assume that β ≤ α . Then we ma y pr o ceed as follo ws. F or i = 1 , . . . , n − 1 put ˆ j i =  j i if i 6 = β α if i = β 23 Then ˆ j 1 , . . . , ˆ j n − 1 are pairwise diﬀeren t and ( ˆ j 1 , . . . , ˆ j n − 1 ; d 1 , . . . , d n − 1 ) satisfy (4.6). Thus b y ind uction h yp othesis there exists a solution ˆ M = ( ˆ m 1 , . . . , ˆ m n ) ∈ F [ t ] ( n − 1) × n for the parameters ( ˆ j 1 , . . . , ˆ j n − 1 ; d 1 , . . . , d n − 1 ). Pu t M =  ˆ m 1 · · · ˆ m α · · · ˆ m n ˆ m α t d n 1 0  ∈ F [ t ] n × ( n +1) , where for j ∈ { 1 , . . . , n − 1 }\{ α } a zero entry o ccurs at th e p ositio n ( n, j ). This matrix do es not y et satisfy (i) –(vii), and w e will tak e care of it later on. Let us ﬁrs t sho w that M is b asic. Computing the n -minors we obtain M ( n +1) = ± t d n ˆ M ( α ) ± ˆ M ( n ) and M ( α ) = ± ˆ M ( n ) whereas for i 6∈ { α, n + 1 } we ha ve M ( i ) = ± t d n ˆ M ( i ) (since the cofactor of the entry 1 in the last row is zero). Hence again t ∤ ˆ M ( n ) together with the b asicness of ˆ M implies the basicness of M . Let u s no w turn to the p rop erties (i) – (vii). First of all it is easy to see that M = ( m ij ) satisﬁes (i), (ii), (iv), and (v). In particular, in the ro w β , wh ere j β = n + 1, we ha ve b y construction deg m β ,n +1 = d β as well as deg m β ,j ≤ d β for j < n + 1. F urthermore, prop erties (vi) an d (vii) are satisﬁed since they are tr ue for ˆ M along with th e facts that β ≤ α and j β = n + 1 > β . Thus, let us turn to (iii). By construction prop ert y (iii) is satisﬁed f or those ind ices i for whic h j i < α . The only obstacle o ccurs w h en j i > α . In this case we also h a ve d eg m ij < d i for j i < j < n + 1, but the en try in the last column do es not n ecessarily satisfy this degree constrain t. Due to prop ert y (iii) for ˆ M we hav e in stead deg m i,n +1 ≤ d i if j i ≥ α . W e will no w p erf orm elemen tary column op erations in order to meet this ﬁnal degree constraint. These column op erations w ill only c hange the last column of M and do not destro y an y of the pr op erties mentio n ed ab o ve. F or l = α + 1 , . . . , n w e consecutiv ely p erform the follo wing steps. If there exists an ind ex i 0 suc h that j i 0 = l and deg m i 0 ,n +1 = d i 0 then w e add a suitable constant multiple of the l -th column of M to the last column such that the resulting en try at p osit ion ( i 0 , n + 1) has degree strictly less than d i 0 . This is p o s s ible since d eg m i 0 ,j i 0 = d i 0 . No w the resulting matrix satisﬁes (iii) for all ind ices i suc h that j i ≤ l . Moreo ver, all other degree constrain ts r emain v alid. In particular, in the ro w β for whic h j β = n + 1 w e still h av e deg m β ,n +1 = d β . T his wa y w e ﬁnally obtain a solution for the p arameters ( j 1 , . . . , j n ; d 1 , . . . , d n ). (c) It remains to consider the case where th ere exists some index β > α suc h that j β = n +1. Notice that β < n . F or i = 1 , . . . , n − 1 p ut ( ˆ j i , ˆ d i ) =  ( j i , d i ) if i 6 = β ( α, d n + d β ) if i = β Then ˆ j 1 , . . . , ˆ j n − 1 are pairwise diﬀerent and ( ˆ j 1 , . . . , ˆ j n − 1 ; ˆ d 1 , . . . , ˆ d n − 1 ) satisﬁes (4.6) s in ce ˆ d β = d n + d β ≥ d n > 0. Thus, b y ind uction hyp othesis th er e is a solution ˆ M = ( ˆ m 1 , . . . , ˆ m n ) ∈ F [ t ] ( n − 1) × n for the parameters ( ˆ j 1 , . . . , ˆ j n − 1 ; ˆ d 1 , . . . , ˆ d n − 1 ). Pu t M =  ˆ m 1 · · · ˆ m α · · · ˆ m n 0 t d n 1 1  ∈ F [ t ] n × ( n +1) , where for i ∈ { 1 , . . . , n − 1 }\{ α } a zero entry o ccurs at the p ositio n ( n, i ). The n -min ors of M are giv en b y M ( n +1) = ± ˆ M ( n ) ± t d n ˆ M ( α ) and M ( n ) = ± ˆ M ( n ) , wh ereas for i < n w e ha ve M ( i ) = ± ˆ M ( i ) . Thus M is b asic. F ur thermore, pr op erties (i) – (vii) are s atisﬁed 24 for all ro ws with in dex i 6 = β , and only the β -th ro w n eeds to b e adjusted. Sin ce ˆ M satisﬁes (iv), (vi), and (vii) the β -th row and the n -th ro w of M are giv en by (0 , . . . , 0 , t d n + d β , 0 , . . . , 0 , f β , f β +1 , . . . , f n , 0) , for some f l ∈ F , (0 , . . . , 0 , t d n , 0 , . . . , 0 , 0 , 0 , . . . , 1 , 1) , resp ectiv ely , wh ere the en tries t d n + d β and t d n app ear in the α -th p osit ion, and the entry f β is in the β -th p osition. Moreo ver, f β = 1 by (iv). No w we see that we ma y subtract t d β times the n -th ro w of M from the β -th row in order to obtain a n ew matrix M ′ where the β -t h ro w is of the form (0 , . . . . . . . . . , 0 , f β , f β +1 , . . . , f n − t d β , − t d β ) , where still th e entry f β is in the β -th p ositio n . S ince j β = n + 1 n o w prop e r ties (i) –(vii) are satisﬁed for i = β , whereas the other rows d id not c h ange. This ﬁnally shows that M ′ satisﬁes all requirements (i) – (vii).  5 The M o diﬁed Ro ok Problem In th is short section w e will brieﬂy discuss Problem 4.5 for k := n − 1 giv en num b ers. First of all, n otice that th e matrix ˆ D in (4.2 ) is the addition table of the group Z n := Z /n Z if the elemen ts are ord ered su itably . This h as actually b een used implicitly in Remark 4.6. The add itiv e group Z n allo ws us to reformulat e the pr oblem. In ord er to do so let P := { ( x 1 , . . . , x n − 1 ) ∈ Z n − 1 n | x 1 , . . . , x n − 1 are p airwise diﬀerent } and S := { r ∈ Z n − 1 n | ∃ x, y ∈ P : r = x + y } . As a consequence, P r oblem 4.5 is s olv able for all r ∈ Z n − 1 n if and only if S = Z n − 1 n . Here are some simp le prop e r ties of the set S . Prop osition 5.1 (i) γ 1 ∈ S for all γ ∈ Z n . (ii) If r ∈ S , then τ ( r ) ∈ S for all p e r m u tations τ in the symmetric group S n − 1 . (iii) If r ∈ S , then γ r ∈ S for all γ ∈ Z × n . (iv) If r ∈ S , then r + γ 1 ∈ S for all γ ∈ Z n . (v) P ⊆ S . Pr o of: Prop erties (i), (ii) and (iii) are ob v ious , whereas (iv) follo w s from the fact that if x ∈ P then x + γ 1 ∈ P f or all γ ∈ Z n . Let us no w turn to (v). If r ∈ P , then the en tries of r attain n − 1 of the n diﬀerent elemen ts in Z n . Using (ii) w e ma y assume that r = (0 , 1 , . . . , α − 1 , α + 1 , . . . , n − 1) for some α ∈ Z n . Again by (ii) w e ha ve r ∈ S ⇐ ⇒ r ′ := ( α + 1 . . . . , n − 1 , 0 , 1 , . . . , α − 1) ∈ S . By (iv) this in tur n is equiv alen t 25 to s := r ′ − α 1 = (1 , 2 , . . . , n − 1) ∈ S . Hence it suﬃ ces to show that s ∈ S . F or n b eing o dd one has s + x = y wher e x = (2 , 3 . . . , n − 1 2 , n +1 2 , n +3 2 , . . . , n − 1 , 0) , y = (3 , 5 , . . . , n − 2 , 0 , 2 , . . . , n − 3 , n − 1) . Since n is o dd x, y are in P which sh o ws that s ∈ S . F or n even one has s + x = y wh ere x = ( n 2 , n 2 + 1 , . . . , n − 2 , n − 1 , 1 , . . . , n 2 − 2 , n 2 − 1 ) , y = ( n 2 + 1 , n 2 + 3 , . . . , n 2 + n − 3 , n 2 + n − 1 , n 2 + 2 , . . . , n 2 + n − 4 , n 2 + n − 2) Again, x, y ∈ P , s ho wing the desired result.  Prop osition 5.2 If r = ( r 1 , . . . , r n − 1 ) ∈ Z n − 1 n has at most t wo d iﬀeren t entries, that is, |{ r 1 , . . . , r n − 1 }| ≤ 2 , then r ∈ S . Pr o of: If r 1 = . . . = r n − 1 , then th e assertion is in Prop ositi on 5.1(i). Otherwise, using Prop osition 5.1(ii) we ma y assume r = ( α, . . . , α, β , . . . , β ) f or some α 6 = β . Using part(iv) of th at pr op osition we m a y ev en assume that α = 0. Thus, let r = (0 , . . . , 0 | {z } n − 1 − f , β , . . . , β | {z } f ) for some 1 ≤ f ≤ n − 2 . In ord er to prov e r ∈ S let l b e the additiv e order of β in Z n and p ut t := ( t 1 , . . . , t n ) := (0 , β , . . . , ( l − 1) β , 1 , 1 + β , . . . , 1 + ( l − 1) β , . . . , β − 1 , 2 β − 1 , . . . , lβ − 1) . That is, the ent r ies of t are sorted according to the group h β i and its cosets. No w pu t x = ( t 1 , . . . , t n − f − 1 , t n − f +1 , . . . , t n ) , y = ( − t 1 , . . . , − t n − f − 1 , β − t n − f +1 , . . . , β − t n ) . Then r = x + y and, obvio u sly , x ∈ P . In order to see that y ∈ P notice ﬁrst th at the ﬁrs t n − f − 1 en tries are ob viously pairwise d iﬀeren t, and so are the last f en tries. Assume no w β − t j = − t i for s ome n − f + 1 ≤ j ≤ n and 1 ≤ i ≤ n − f − 1. Th en t j = β + t i . But b y construction β + t i = t i +1 if i 6∈ l Z and β + t i = t ( m − 1) l +1 if i = ml . Sin ce j > n − f > i this shows that β + t i 6 = t j . Hence y ∈ P and thus r ∈ S .  Notice th at the v ector t ab ov e could also b e deﬁn ed acc ord ing to a diﬀeren t ordering of the cosets of h β i . T his sho ws, th at th ere are man y wa y s of writing r = x + y for s ome x, y ∈ P . Unfortunately , w e are not a ware of any w ay to ge n eralize the last pro of to v ectors r ∈ Z n − 1 n with 3 or more d iﬀeren t entries. 6 Extension to General Automorphisms — An Example So far we ha ve stud ied σ -CC C’s in F [ z ] n where n | ( q − 1) and where th e automorph ism σ induces a cycle of length n on the primitiv e idemp ote nts of A . In this section we will 26 brieﬂy illustrate how the results can b e utilized for general automorphisms if n | ( q − 1). F or ease of notation let u s restrict to the follo wing example. Let q = 8 and n = 7. Consider the automorphism σ ∈ Aut F ( A ) deﬁned by σ ( e 1 ) = e 2 , σ ( e 2 ) = e 3 , σ ( e 3 ) = e 1 , σ ( e 4 ) = e 5 , σ ( e 5 ) = e 6 , σ ( e 6 ) = e 7 , σ ( e 7 ) = e 4 . In cycle n otation this reads as ( e 1 , e 2 , e 3 )( e 4 , e 5 , e 6 , e 7 ). Deﬁne A 1 := F × F × F and A 2 := F × F × F × F and denote the p rimitiv e id emp oten ts of A 1 (resp. A 2 ) simply by e 1 , e 2 , e 3 (resp. e 4 , e 5 , e 6 , e 7 ). T h en it is straigh tforward to establish the isomorph ism A [ z ; σ ] 7− → A 1 [ z ; σ 1 ] × A 2 [ z ; σ 2 ] , g − →  g (1) + g (2) + g (3) , g (4) + g (5) + g (6) + g (7)  , (6.1) where for i = 1 , 2 the automorphism σ i on A i is deﬁned by the cycle ( e 1 , e 2 , e 3 ) and ( e 4 , e 5 , e 6 , e 7 ), resp ect ively . F urthermore, a p olynomial in A [ z ; σ ] is (semi-)reduced if and only if eac h factor in A i [ z ; σ i ] is (semi-)reduced. As a consequence, the in v estigation of left ideals and direct su mmands in A [ z ; σ ] amounts to the study of th e same t yp e of ob jec ts in the rings A i [ z ; σ i ]. Since for these rings the automorphism ind uces a cycle of maximal length on the primitiv e idemp oten ts this brings u s to the situatio n of the previous sections. Notice, h o we ver, that for the ring A 1 [ z ; σ 1 ] the length n 1 := 3 is not a divisor of q − 1, and th us , A 1 6 ∼ = F [ x ] / h x 3 − 1 i . I n Remark 3.15 w e men tioned that one can p ro ve the results of S ection 3 in this case as wel l w ith the only exception of Theorem 3.5(4) wh ic h do es not mak e sense anymore. Along w ith the isomorp hism (6.1) this is s uﬃcien t in order to construct CCC’s f or this au tomorp h ism as well . Let us illustrate this idea by an example. Example 6.1 L et α ∈ F 8 b e the p r imitiv e elemen t satisfying the identit y α 3 + α + 1 = 0. F or i = 1 , 2 let ξ i : A i [ z ; σ i ] − → M i b e th e isomorphism w ith the according matrix r ing as introd uced in Prop osit ion 3.2. The matrices M 1 =   0 0 0 0 1 α 4 0 0 0   ∈ M 1 and M 2 =     α 6 1 α 0 0 0 0 0 0 0 α 3 1 0 0 0 0     ∈ M 2 are obviously b asic and semi-reduced, and thus so are th e p olynomial s g 1 = ξ − 1 1 ( M 1 ) = e 2 + z α 4 e 3 ∈ A 1 [ z ; σ 1 ] and g 2 = ξ − 1 2 ( M 2 ) = α 6 e 4 + α 3 e 6 + z ( e 5 + e 7 ) + z 2 αe 6 ∈ A 2 [ z ; σ 2 ]. Using the isomorphism in (6.1) w e obtain the s emi-red u ced and basic p olynomial g = g 1 + g 2 = e 2 + α 6 e 4 + α 3 e 6 + z ( α 4 e 3 + e 5 + e 7 ) + z 2 αe 6 ∈ A [ z ; σ ] . Its supp o r t is giv en b y T g = { 2 , 4 , 6 } . As a consequence, • h g i is a direct sum mand of rank 3 of the left F [ z ]- mo dule A [ z ; σ ]. No w [8, Thm. 7.13(b)] (whic h is also v alid for semi-reduced p olynomials) tells us that the matrix G :=   p − 1 ( g (2) ) p − 1 ( g (4) ) p − 1 ( g (6) )   ∈ F [ z ] 3 × 7 is a minimal enco der for the σ - C CC C := p − 1 ( • h g i ) ⊆ F [ z ] 7 . By construction, the co d e has F orney in dices 1 , 2 , 1. Of course, basicness and min imalit y of the matrix G can also b e c hec ked directly once th e rows of the m atrix ha ve b een computed using the mapp ing p from (2.4). Using a computer alge b ra routine one ﬁ nds that the distance of C is 12. In other words, the co d e attains th e Griesmer b o u nd, see [6, Eq. (1.3)]. 27 Concluding Remarks In this pap er w e stud ied a particular class of CCC’s. W e show ed that th e existence of suc h codes with an y giv en algebraic paramete r s can b e redu ced to solving a certain com binatorial problem. Under the assumption this problem is solv able f or all p o ssib le instances this shows that the class of σ -CCC’s is, in a certain sense, as r ic h as the class of all CC’s. W e strongly b eliev e that the com bin atorial problem is solv able for all instances, but that has to r emain op e n for futur e r esearc h. Moreo ve r , the p oten tial of our approac h needs to b e f u rther exploited with resp ect to err or-correcting prop erties. 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A Matrix Ring Description for Cyclic Convolutional Codes

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