On linear completely regular codes with covering radius $rho=1$. Construction and classification

On linear completely regular co des with co v ering radius ρ = 1. Construction and classifi ca t i on ∗ J. Borges, J. Rif` a † , V.A. Zino viev ‡ No vem b er 5, 201 8 Abstract Completely regular codes with cov ering radius ρ = 1 m ust hav e mini- mum distance d ≤ 3. F or d = 3, such co des are perfect and their parame- ters are w ell known. In this p ap er, the cases d = 1 and d = 2 are stu died and completely characteri zed when the co des are linear. Moreo ver, it is prov en that all these co des are completely transitiv e. Keywords: Linear completely regular codes, completely transitive co des, cov ering radius. 1 In tro duction and Bac kground Let F q = GF ( q ) b e the Galois Field with q elements, wher e q is a prime pow e r. F n q denotes the n -dimensional vector space over F q . The all-zero v ector in F n q is denoted by 0 . Let wt( v ) denote the Hamming weight o f a vector v ∈ F q ∗ This w ork has been partially supported b y the Spanish MICINN Grants MTM 2006-03250, TSI2006-1400 5-C02-01 and PCI2006-A7-0616, the A GAUR grant 2008PIV00050 and also b y the Russian fund of fundamen tal r esearc hes 06-01-00226. † J. Borges and J. Rif` a are with the Departmen t of Information and C ommu- nications Engineering, Unive rsitat Aut` onoma de Barcelona, 0819 3-Bellaterra, Spain (email: { joaquim.b orges,josep.rif a } @autono ma.edu). ‡ V.A. Zino viev is with the Institute for Pr oblems of Information T ransmission of the Rus- sian Academ y of Sciences, Bol’ shoi Karetn yi per. 19, GSP-4, Moscow, 127994, Russia (e-mail: zino v@iitp.ru). 1 (i.e. the n umber o f its nonzer o p ositions), a nd d ( v , u ) = wt( v − u ) deno tes the Hamming distanc e b etw een tw o vectors v and u . Given v ∈ F n q , deno te by supp ( v ) the supp ort of the vector v , that is, the set of coo rdinate p ositions where v has nonzero entries. W e say that a vector u = ( u 1 , . . . , u n ) ∈ F n q c overs a vector v = ( v 1 , . . . , v n ) ∈ F n q if v i 6 = 0 implies v i = u i . A q -ary c o de C of length n is a subset of F n q . If C is a k -dimensional linear subspace of F n q , then C is a line ar co de, denoted by [ n, k , d ] q , where d is the minimum distanc e betw ee n any pair of co de words. Let C b e a q -ary co de with minimum distance d , the p acking ra dius of C is e =  d − 1 2  . Such a co de is said to be an e - err or-c orr e cting co de. Given any v ecto r v ∈ F n q , its distanc e to the co de C is d ( v , C ) = min x ∈ C { d ( v , x ) } and the c overing r adius o f the co de C is ρ = max v ∈ F n q { d ( v , C ) } . Clearly e ≤ ρ and C is s aid to b e p erfe ct when e = ρ . F or any x ∈ F n q , let D = C + x b e a tr anslate of C . The weight wt( D ) of D is the minim um weigh t of the co dewords of D . Definition 1.1 A q - ary c o de C is c al le d co mpletely reg ular if the weight dis- tribution of any tr anslate D of C is u niquely define d by the weight of D . Equiv alently , C is completely reg ular if for all x ∈ F n q such that d ( x , C ) = t , the n umber of co dewords at distance i (0 ≤ i ≤ n ) from x depends only on t and i . Given a co de C with covering radius ρ , let C ( ρ ) be the set of vectors at distance ρ from C . The next statement can be found in [7] for binar y co des. F or the non-binary cas e it can be prov en in similar wa y . 2 Lemma 1 .2 If a q -ary c o de C is c ompletely r e gular with c overing r adius ρ , then C ( ρ ) is also c ompletely re gular. A line ar automorphism o f F n q is a co or dinate permutation together with a pro duct by a nonzer o scala r v alue at eac h p os itio n. Such an automorphism σ can be represented b y a n × n monomial matrix M such that x M = σ ( x ), for all x ∈ F n q . F rom now on, if C ⊆ F n q is a linear co de, the ful l aut omorphism gr oup of C , denoted Aut( C ), is the g r oup of linear automorphisms o f F n q that leav es C inv ar ia nt. W e sa y that Aut( C ) is tr ansitive if it is transitiv e when acts on the set of weight one vectors of F n q . Lemma 1 .3 L et C, D ⊆ F n q b e two line ar e qu ivalent c o des, i.e. ther e is a line ar automorphism σ of F n q such that D = σ ( C ) . Then Aut( C ) is tr ansitive if and only if Aut( D ) is tra nsitive. Pro of: Notice that for all g ∈ Aut ( C ), σ g σ − 1 ∈ Aut( D ). Assume that Aut( C ) is transitive. Let x and y b e weight one v ector s, w e w ant to find δ ∈ Aut( D ) s uch that δ ( x ) = y . Let τ ∈ Aut( C ) such that τ ( σ − 1 ( x )) = σ − 1 ( y ), then σ τ σ − 1 ( x ) = y and σ τ σ − 1 ∈ Aut( D ). The statement then follows r e versing the ro les of C and D .  F or a linear co de C , the gr oup Aut( C ) a cts on the set o f co sets o f C in the following wa y: for all φ ∈ Aut( C ) and for every vector v ∈ F n q we hav e φ ( v + C ) = φ ( v ) + C . In [5] and [10] the following definition has b een giv en for the cas e of linear co des. Definition 1.4 L et C b e a q -ary line ar c o de with c overing r adius ρ . Then C is c ompletely tr ansitive if Aut( C ) has ρ + 1 orbits when acts on the c osets of C . Since t wo cosets in the same orbit should hav e the same w eig ht distribution, it is clear that any co mpletely tra nsitive co de is co mpletely reg ular. The fol- lowing statement can b e genera lized for the case ρ > 1 replacing transitivity by ρ -homogeneity [10]. Here, w e are only interested in the case ρ = 1. 3 Lemma 1 .5 L et C b e a [ n, k , d ] q c o de with c overing r adius ρ = 1 . If Aut( C ) is tr ansitive, then C is c ompletely t r ansitive. Pro of: O bvious, since all cosets of C , different of C , hav e leaders of weigh t 1. Thus, all suc h cosets a re in the same or bit.  It has b een conjectured [7] for a long time that if C is a completely regula r co de and | C | > 2, then e ≤ 3. F or the sp ecia l ca s e of binary linear completely transitive co des [1 0], the pro blem of existence is solved: it is proven in [2, 3] that for e ≥ 4 such no nt rivial co des do not exist. The co njecture is als o prov en for the case o f p erfect c o des ( e = ρ ) [12, 14] and quasi-per fect ( e + 1 = ρ ) uniformly pack ed co des [6, 13], defined and s tudied also in [1, 11]. When e ≤ 3, there are w ell known completely regula r c o des a nd, recently , we ha ve presented new constructions o f binary and no n- binary co mpletely reg- ular co des [4, 8, 9]. How ever, ther e do es no t exist a gener al cla ssification of completely r egular co des with e ≤ 3. In this pa p e r we consider q -ary linea r completely regular co des with ρ = 1 . A sur pr ising fact is that to characterize all linear c o mpletely regula r co des with ρ = 1 we nee d o nly three construc- tions ( q - r ep eated co de construction, direct construction a nd Kroneck er pro duct construction). The pap er is org anized as follows. In Section 2 w e pres e n t the q times rep eating construction to obtain linear or nonlinear q -ary completely r egular co des with d = 1. In Section 3, we g ive a direct construction to obtain q -ary linear completely regula r co des with ρ = 1 and d ∈ { 1 , 2 } , we also int ro duce the Kronecker pro duct of ma trices as an imp orta n t to ol to characterize q -ary linear completely re g ular co des with ρ = 1 a nd, finally , we show that all such completely reg ular co des a re completely transitive too. 4 2 Completely regular co des with d = 1 and the q -r ep eated co de cons truction W e start with a first example of family of completely regula r codes with mini- m um distance d = 1. Lemma 2 .1 L et C b e a p erfe ct (binary or non-binary) c o de. Then C ( ρ ) has minimum distanc e 1. Pro of: Without loss o f gener ality , w e assume that 0 ∈ C . Let x ∈ C ( ρ ) with wt( x ) = ρ and let x ′ be a vector such that d ( x , x ′ ) = 1 and wt( x ′ ) ≥ ρ . W e claim that x ′ ∈ C ( ρ ) and then the minim um distance in C ( ρ ) is 1. Assume to the contrary that x ′ / ∈ C ( ρ ), then clearly d ( x ′ , C ) = ρ − 1. Notice also that a co deword y at distance ρ − 1 of x ′ cannot b e 0 . Hence we obtain a contradiction bec ause x is at dis tance ρ from more than one co deword.  As we hav e s e e n in Le mma s 1.2 and 2.1, the co vering set C ( ρ ) of any p erfect co de is a completely regula r co de with minim um distanc e d = 1. In pa rticular, if C is a single er ror-co rrecting co de ( e = 1), then C ( ρ ) is exactly the complemen t of C . But these a re not the o nly examples of completely regular co des with d = 1. Let C be a [ n, k , d ] q co de. W e construct the q -r epe ated co de C ′ ⊆ F n +1 q of C a s follows: for an y co deword x = ( x 1 , . . . , x n ) ∈ C , w e hav e q co dewords in C ′ , namely (0 , x 1 , . . . , x n ) , (1 , x 1 , . . . , x n ) , . . . , ( q − 1 , x 1 , . . . , x n ) . Lemma 2 .2 L et C b e a [ n, k , d ] q c o de and let C ′ ⊆ F n +1 q b e its q -r ep e ate d c o de. L et x = ( x 1 , . . . , x n ) ∈ F n q b e a ve ctor at distanc e i fr om α i c o dewor ds in C and at distanc e i − 1 fr om α i − 1 c o dewor ds in C . Th en, any ve ctor of the form x ′ = ( x 0 , x 1 , . . . , x n ) is at distanc e i fr om exactly α i + ( q − 1) α i − 1 c o dewor ds in C ′ . Pro of: F or any co deword z = ( z 1 , . . . , z n ) ∈ C s uch tha t d ( z , x ) = i we hav e that z ′ = ( x 0 , z 1 , . . . , z n ) is in C ′ and d ( z ′ , x ′ ) = i . Mo reov e r , for an y co deword 5 y = ( y 1 , . . . , y n ) ∈ C such tha t d ( y , x ) = i − 1, w e hav e that the q − 1 vectors of the for m ( y 0 , y 1 , . . . , y n ) with y 0 6 = x 0 are codewords in C ′ and they a re at distance i from x ′ . It is c lear tha t there are no mo re co dewords in C ′ at distance i from x ′ .  Theorem 2 .3 ( q -R ep e ate d c o de c onstruction ) L et C b e a [ n, k , d ] q c o de with c overing r adius ρ . Then the q -r ep e ate d c o de C ′ ⊆ F n +1 q has ρ ′ = ρ and minimum distanc e d ′ = 1 . Mor e over C ′ is c ompletely r e gular if and only if C is c ompletely r e gular. Pro of: F or any v e ctor x ′ = ( x 0 , x 1 , . . . , x n ) ∈ F n +1 q , ca ll x = ( x 1 , . . . , x n ) ∈ F n q the corresp onding ‘reduced’ v ector. Suppose that y = ( y 1 , . . . , y n ) ∈ C is a co deword at minim um distance from x . Then it is clear that y ′ = ( x 0 , y 1 , . . . , y n ) is a co deword in C ′ at minimum distance fr o m x ′ . The r efore ρ = ρ ′ . Now, assume that C is completely regula r. F or any vector x = ( x 1 , . . . , x n ) ∈ F n q at distance t ≤ ρ from C , define α i ( t ) as the num b er of co dewords in C at distance i from x (0 ≤ i ≤ n ). As C is completely r e gular, we know that α i ( t ) do es no t dep end on x , but just on t and i . W e w ant to see that for the vector x ′ = ( x 0 , x 1 , . . . , x n ) ∈ F n +1 q , which is at distanc e t fo rm C ′ , we also have that the n umber of co dewords in C ′ at dista nce i , say α ′ i ( t ), depends only on t a nd i . But this is s traightforw ard b ecause using Lemma 2.2 we hav e α ′ i ( t ) = α i ( t ) + ( q − 1) α i − 1 ( t ), for all i = 0 , . . . , n , a nd α ′ n +1 ( t ) = ( q − 1) α n ( t ). Conv ers ely , assume that C is not completely r e gular. Let x , y ∈ F n q be such that d ( x , C ) = d ( y , C ) = t > 0 and let α x ,i ( t ) (resp ectively α y ,i ( t )) denote the nu mber of codewords at dis ta nce i from x (r e s pe c t. y ), for 0 ≤ i ≤ n . Since C is not completely r egular, w e can select x and y such that α x ,i ( t ) 6 = α y ,i ( t ) for some i ≥ t . L e t i b e the minimum po ssible such v alue (p ossibly , i = t ), that is α x ,i − 1 ( t ) = α y ,i − 1 ( t ). Then, for the q - r ep eated v ectors x ′ and y ′ , we have α ′ x ′ ,i ( t ) 6 = α y ′ ,i ( t ) b y Lemma 2.2. Conse quent ly , C ′ is not completely reg ula r.  Hence, w e can star t with any completely regular code and obtain an infinite family o f completely regular co des with the same cov ering radius. W e remark 6 that this construction is also v alid for nonlinear co des . Conv ers ely , for the linear cas e with d = 1, w e hav e the following: Corollary 2 . 4 L et C 6 = F n q b e a q -ary line ar c o de with minimum distanc e d = 1 and c overing r adius ρ . Then C c an b e obtaine d u s ing t he q -r ep e ate d c o de c onstruction (r ep e ating the pr o c ess some numb er of times) fr om a c o de D which has minimum distanc e gr e ater than one and c overing r adius ρ . Mor e over, C is c ompletely r e gular if and only if D is c ompletely r e gular. Pro of: Let G b e a generator matrix for C containing all linear indep endent co dewords of weigh t 1. The desired co de D is then o btained removing from G all r ow v ectors of weight 1 and the resulting zero c olumns. As we hav e se e n in Theorem 2.3, the cov e r ing ra dius do es not change a nd C is completely regular if and o nly if D is co mpletely regula r.  3 Completely regular co des wit h ρ = 1 Since e ≤ ρ , completely regular co des with ρ = 1 must hav e minim um distance d ≤ 3. When d = 1 we have seen, in the previous section, that we can obtain these codes using the q -r ep eated cons truction starting fro m co des with the s ame cov er ing r adius ρ = 1 and with minimum dista nce g reater tha n 1. F or d = 3, we ha ve e = ρ and thes e co des are p erfect. Linea r p erfect co des with e = 1 are the well known Hamming co des. Therefore, if ρ = 1 the case to fo cus our interest is d = 2 . A first example where we constr uct linear c o des with these par ameters, ρ = 1 a nd d = 2 , is given b y the following theorem. Theorem 3 .1 ( D ir e ct c onst ru ction ) L et C b e a [ m + 1 , m, d ] q c o de define d by a gener ating matrix G , G = [ I | h ] , wher e I is the identity matrix of or der m , and h is an arbitr ary nonzer o c olumn ve ctor fr om F m q . Then, if wt( h ) < m , t he c o de C is a c ompletely r e gular c o de 7 with d = ρ = 1 . If wt( h ) = m , then C is a c ompletely r e gular c o de with d = 2 and ρ = 1 . Pro of: Clearly , if wt( h ) < m , then the minim um dis tance of C is 1 and if wt( h ) = m , then the minim um distance is 2. A parity chec k matrix for C is given b y H = [ − h t | 1] and any pair of columns ar e linearly dep endent. Hence ρ = 1. In order to see that C is completely regular, w e take a vector x at distance 1 from C (or , the same, x / ∈ C ) and we pro ve that the num b er o f co dewords at distance 1 fro m x is alwa ys the same. Assume, witho ut lo ss of g enerality , that x = ( x 1 , . . . , x m +1 ) ha s w eight 1 . Let w = wt( h ) and let x i be the nonzero co ordinate of x . First, w e consider the case i < m +1 . The co dewords at distance 1 from x ar e 0 , x i v ( i ) , where v ( i ) is the i -th row of G (notice that v ( i ) has weight 2, otherwise x w ould be a co deword) and the co dewords of w e ig ht 2 with the v alue x i at the i -th coor dinate which ar e of the form: y ( ij ) = x i v ( i ) + α j v ( j ) for all r ow v ectors v ( j ) of weight 2 ( j 6 = i ), where α j ∈ F q is taken s uch that the last co ordinate of y ( ij ) is zero. Thus, we hav e w + 1 co dewords at distance 1 from x . Fina lly , consider the cas e i = m + 1. The co dewords at distance 1 from x are 0 a nd the w co dewords of the form y ( j ) = α j v ( j ) , where v ( j ) has weigh t 2 and α j ∈ F q is taken such that the last co ordinate of y ( j ) is x i . Ag ain, we obtain w + 1 co dewords at dis tance 1 fr om x .  F rom now on, our goa l is to classify a ll the linear completely regular c o des with ρ = 1 and d = 2. W e will b egin by introducing the Kro neck er pro duct of matric es and showing that this tool will help us in the construction of linear completely regular codes with the required parameter s. Definition 3.2 The Kr one cker pr o duct of two matric es A = [ a r,s ] and B = [ b i,j ] over F q is a new matr ix H = A ⊗ B obtaine d by changing any element a r,s in A by the matrix a r,s B . 8 A r ep et ition c o de is a [ n, 1 , n ] q co de. In this pap er, we assume that such a rep etition co de has all codewords of the form ( c, c, . . . , c ) for c ∈ F q . Lemma 3 .3 L et H b e [ n , k , 3] q Hamming c o de. Then, Aut ( H ) is tr ansitive. Pro of: Let G and H b e g enerator and par ity chec k matr ic e s, resp ectively , for H . Let x a nd y be a n arbitrar y pair of weight one vectors. W e wan t to find a linear automorphism of H that se nds x to y . It is straightforward to find an in vertible ( n − k ) × ( n − k ) matrix K , with entries in F q and s uch that K H x t = H y t . Since H is a parity chec k matrix of a Hamming co de, there exists a monomial n × n matrix M such that K H = H M t . Since H ( M t G t ) = K H G t = 0, w e ha ve that GM is also a g enerator matrix for H . Thus, M is the monomial matr ix asso c ia ted to a linear automorphism φ ∈ Aut( H ). Now, K H x t = H y t implies H M t x t = H y t . As M t x t and y t hav e weight one and H has no r ep eated columns, we conclude x M = y or , the same, φ ( x ) = y .  Theorem 3 .4 Le t C b e the line ar c o de over F q which has H = A ⊗ B as a p arity che ck matrix, wher e A is a gener ator matrix for the r ep etition [ n a , 1 , n a ] q c o de of length n a and B is a p arity che ck m atr ix of a H amming c o de with p ar ameters [ n b , k b , 3] q , wher e n b = ( q m b − 1 ) / ( q − 1) and k b = n b − m b . (i) Co de C has length n = n a · n b , dimension k = n − m b and c overing r adius ρ = 1 . (ii) If n a > 1 , then the minimum distanc e of C is d = 2 . If n a = 1 , then d = 3 . (iii) Aut( C ) is tr ansitive and, ther efor e, C is a c ompletely tr ansitive c o de and a c ompletely r e gular c o de. Pro of: It is straightforw a rd to check that the c o de C has length n = n a · n b , dimension k = n − m b and cov ering radius ρ = 1 . If n a = 1, then C is a Hamming co de a nd d = 3. If n a > 1, then H has rep eated columns and d = 2 . 9 The matr ix H is of the form H = [ B B · · · B ] , where B is a parity chec k matrix for a Hamming c o de H . By Lemma 3.3, Aut( H ) is transitive on the set o f weight one v ectors with suppor t co ntained in the set of co ordinate po sitions of H . Hence we have that Aut( C ) is trans itive on each s et of w eig ht one v ectors with suppor t co nt ained in the set o f n b co ordinate p ositions corres p o nding to each submatrix B . Now cons ider tw o vectors of weight one x ∈ F n q and y ∈ F n q , suc h that the nonzero entry of x is x ∈ F q at position i a nd the nonzero en try of y is y at p os ition j and assume tha t i and j are in different n b -sets, i.e. sets o f car dinality n b , o f c o ordinate p ositions . Let ϕ ∈ Aut ( C ) suc h that ϕ ( x ) = x ′ , where x ′ has w eight one with its nonzero en try equal to y a t po sition i ′ , in the same n b -set of co or dina te positions, where the co lumn v ector of H in p os itio n i ′ is the same that the column vector in p ositio n j . Clearly , the transp osition τ = ( i ′ , j ) is a linear automorphism of C . Thus, τ ( ϕ ( x )) = y . Therefore, w e hav e prov en that Aut( C ) is transitive and, by Lemma 1.5, C is a co mpletely tra nsitive code and hence a completely regular co de.  The follo wing step is to prov e that, vice versa, co des constructed in Theo- rem 3.4 are the unique linear co mpletely regula r co des with d = 2 a nd ρ = 1. Lemma 3 .5 L et C b e a c ompletely r e gular [ n, k , 2] q c o de with c overing r adius ρ = 1 . L et n a b e the n umb er of c o dewor ds at distanc e 1 fr om any ve ct or x / ∈ C . Then the fol lowing statements ar e e quivalent: (i) F or any p air of c o or dinate p ositions i and j , ther e exists a c o dewor d of weight 2 with supp ort { i , j } . (ii) n a = n . (iii) C is a q -ary p art of the whole sp ac e, i.e. | C | = q n − 1 . (iv) Co de C has a gener ator matrix of the form G = [ I | h ] , 10 wher e I is the identity matrix of or der n − 1 , and h is a c olumn ve ctor of weight n − 1 fr om F n − 1 q . (v) Dual c o de of C is e quivalent to a r ep etition [ n, 1 , n ] q c o de. Pro of: Let x / ∈ C , without loss of genera lit y we assume that x has weigh t 1 and let x i be the nonzero coor dinate o f x . Then the co dewords at distance 1 fro m x a r e the all- zero co deword and all co dewords of weigh t 2 with x i at the i - th co ordinate. Such co de words hav e the remaining no nzero co o rdinate in different plac e s (otherwise C would hav e co dewords of weight one). There are n − 1 p ossible differen t places. Hence (i) and (ii) are equiv alent. Define the following simple bipartite graph with vertices which are all p oints of F n q and with edges , connecting the p oints of C with the points C ( ρ ) = F n q \ C , if these tw o p o ints ar e at distance one from eac h other . Coun t the n umber of edges in tw o ways. F r om one side, any co deword o f C is at distance 1 fro m ( q − 1) n po ints of C ( ρ ). F rom the other side, any p oint of C ( ρ ) is at distance 1 from n a po int s of C . Since these n um be rs should b e equa l, we conclude that n ( q − 1) | C | = n a | F n q \ C | , which giv es ( q − 1) n = ( q n − k − 1 ) n a , (1) where k is the dimension of C . It is clea r that n a = n if and only if k = n − 1. This gives the eq uiv alence b etw een (ii) and (iii). The equiv alence b etw een (iii) and (iv), and betw een (iv) a nd (v) are trivial.  Lemma 3 .6 L et C b e a c ompletely r e gular [ n, k , 2] q c o de with c overing r adius ρ = 1 . L et n a b e the numb er of c o dewor ds at distanc e one fr om any ve ctor not in C . If k < n − 1 , then the set of c o or dinate p ositions { 1 , . . . , n } c an b e p artitione d into n a -sets, X 1 , . . . , X n/n a , such that any c o dewor d of weight 2 has its supp ort c ontaine d in one of these sets. 11 Pro of: First note that n a ≥ 2, otherwise C would be a perfect co de with d = 2 which do es not exist. By Lemma 3.5, since k < n − 1, w e a lso hav e n a < n and clear ly n a divides n b y (1). Now, for any vector u / ∈ C of w eight 1, consider the union of the suppor ts of the n a − 1 co dewords of weigh t 2 that cov er u . Denote by X ( u ) such set of co ordinate po sitions and no te that | X ( u ) | = n a . Let v b e ano ther vector o f weigh t 1 such that its supp or t is not in X ( u ). It s uffice s to prov e tha t X ( u ) and X ( v ) are disjoin sets. Assume to the co ntrary that a coor dinate p osition i belo ngs to X ( u ) ∩ X ( v ). This means tha t there is a co deword x of w eight 2 cov er ing u and a co deword y of w eight 2 cov ering v and supp ( x ) ∩ supp ( y ) = { i } . Let y ′ be a multiple o f y such that y ′ i = x i . Then, the co deword z = x − y ′ cov er s u but supp ( z ) * X ( u ) which is a cont radiction.  Corollary 3 . 7 With t he same hyp othesis of L emma 3.6, let D i b e the c o de that has the c o dewor ds of C such t hat t heir s u pp orts ar e c ontaine d in X i and deleting the c o or dinate p ositions outs ide of X i . Then, D i is a line ar c ompletely r e gular c o de of length n a , dimension n a − 1 , minimum distanc e d = 2 , and c overing r adius ρ = 1 . A gener ator matrix for D i is: G i = [ I | h ] , wher e h is a c olumn ve ctor of weight n a − 1 . Pro of: F or any i = 1 , . . . , n/n a , it is s traightforw ard to see that D i is a linear co de of length n a and minimum dista nce d = 2. Moreov er, let Z b e the set o f w eig ht t wo co dewords cov ering some fixed vector of w e ight one. Then Z is a set o f n a − 1 linear indepe nden t co dewords. Thus, b y Theor em 3.1, co de D i is completely regular with ρ = 1.  Now, it is clear that any linear completely r e gular co de C with d = 2 and ρ = 1 can be ‘decomp osed’ into completely regula r co des D i of t y pe ‘direc t construction’. In or der to co mplete the cla s sification we need the following techn ical re s ults. 12 Lemma 3 .8 With the same hyp othesis as in L emma 3.6 , let x = ( x 1 , . . . , x n ) ∈ C and let X j b e one of the sets as in L emma 3.6, such that supp ( x ) ∩ X j 6 = ∅ . Then ther e exists a c o dewor d x ′ = ( x ′ 1 . . . , x ′ n ) ∈ C which c oincides with x in al l p ositions outside of X j , such that | supp ( x ′ ) ∩ X j | ≤ 1 , and wher e for the c ase | supp ( x ′ ) ∩ X j | = 1 , the nonzer o element of x ′ o c cur in any p osition of X j , i.e. for any i j ∈ X j ther e is a such ve ct or x ′ with nonzer o element in p osition i j . Pro of: Let x = ( x 1 , . . . , x n ) ∈ C a nd le t X j be such that supp ( x ) ∩ X j 6 = ∅ . Now, adding co dewords of weigh t 2 with supp or t only in X j (see Lemma 3.6), from x w e easily arrive to x ′ , which has either all zero co ordinates on X j , or exactly one nonzero co or dinate which migh t b e placed on any position of X j .  Prop ositio n 3.9 With t he same hyp othesis as in Le mma 3.6, for e ach j = 1 , . . . , n/n a , take and fix a c o or dinate p osition i j ∈ X j . Le t D ′ b e the c o de that has al l c o dewor ds in C having t heir supp orts c ontaine d in I = { i 1 , . . . , i n/n a } . L et D b e the c o de obtaine d fr om D ′ by deleting al l c o or dinates outs ide of I . Then n/n a ≥ 3 and D is a Hamming c o de of lengt h n/n a . Pro of: Clea rly D is a linear c o de of length n/n a . By Lemma 3.6, since we are as s uming k < n − 1, D is not empty and the minimum weight of D is 3. Th us, we only need to prov e that the covering radius of D is 1. Otherwis e, assume that v is a vector (with co o rdinates in I ) at distance 2 from D . Without loss of generality , we can a ssume that v has weight 2 with supp ( v ) = { i r , i s } , ( i r ∈ X r , i s ∈ X s , r 6 = s ). The cov er ing radius of C is ρ = 1, so we can take x ∈ C at dista nce one from v ′ , where v ′ is the extensio n of vector v adding zero es in all co ordinate pos itions of { 1 , . . . , n } \ I . By Lemma 3 .6, x cannot hav e neither w eig h t 2 nor w eight 1, since the minimum distance o f C is 2. Thus, x is a c o deword of weight 3 with supp ( x ) = { i r , i s , i } . Note that i cannot be in X r or X s , otherwis e, using Lemma 3.8 we could obtain a co deword o f w eight 2 with suppo rt { i r , i s } , contradicting Lemma 3 .6. W e conclude tha t n/n a ≥ 3. Le t i ∈ X t , wher e r 6 = t 6 = s . Again, using Lemma 3.8, we can obtain a co deword 13 x ′ ∈ C such that supp ( x ′ ) = { i r , i s , i t } , x ′ i r = x i r and x ′ i s = x i s . Clea rly , x ′ restricted to the I coordina tes is a co deword in D of weight 3 and cov ers v . Therefore v is not at dista nc e 2 fro m D .  Corollary 3 . 10 Le t C b e a [ n, k , 2] q c ompletely r e gular c o de with c overing r a- dius ρ = 1 and let n a b e the numb er of c o dewor ds at distanc e one fr om any ve ctor not in C . Then, either n a = n , k = n − 1 and C has gener ator matr ix : G = [ G 1 ]; or C has gener ator matrix: G =         G 1 0 0 0 . . . 0 0 0 G n/n a M 1 · · · M n/n a         , (2) wher e G i is a gener ator matrix of a [ n a , n a − 1 , 2] q c o de ( which is c ompletely r e gular) for al l i = 1 , . . . , n/ n a , and M i has n a − 1 zer o c olumns and one c olumn h i such that h h 1 · · · h n/n a i is a gener ator matrix of a Hamming c o de H . Pro of: W e have a lready seen in Theorem 3 .1 the case n a = n , k = n − 1. Now, let n a < n . By Corollar y 3.7 and Prop osition 3.9, it is clear that co de C ′ generated by G is a sub co de of C . But, the n umber of rows (whic h ar e all linear indep endent) of G is: n n a · ( n a − 1 ) + dim ( H ) . Since (1), the length of H is n n a = q n − k − 1 q − 1 , the dimension of H is ( n/n a ) − ( n − k ). Therefor e dim ( C ′ ) = n n a · ( n a − 1 ) + n n a − n + k = k . Hence, dim ( C ′ ) = dim ( C ) and co nsequently C ′ = C .  14 Prop ositio n 3.11 Le t C b e a [ n, k , 2] q c ompletely r e gular c o de with c overing r adius ρ = 1 . L et n a b e the numb er of c o dewor ds at distanc e one fr om any ve ctor not in C . L et A b e a gener ator matrix of a r ep etition [ n a , 1 , n a ] -c o de and let B b e a p arity che ck matrix for a Hamming q -ary c o de of length n b = n/n a . (i) If k = n − 1 , t hen c o de C is e qu ivalent t o a c o de with p arity che ck matrix H = A . (ii) If k < n − 1 , then C is e quivalent to a c o de with p arity ch e ck matrix H = A ⊗ B . Pro of: If k = n − 1, b y Cor ollary 3.10, co de C is given by a g enerator matrix of a [ n a , n a − 1 , 2] q co de. A parity chec k matrix for an equiv a lent co de to C is the gener ator matrix of a rep etition [ n a , 1 , n a ]-co de. If k < n − 1, we can start with a gene r ator matr ix as in (2). Then, we m ultiply the first n a ( n a − 1) rows by appropriate v alues. After, we c an multiply the columns to obtain the following generator matr ix : G =         G ′ 0 0 0 . . . 0 0 0 G ′ M 1 · · · M n/n a         , (3) where G ′ is a ( n a − 1 ) × n a matrix G ′ = [ I | h ] . Up to equiv alence, w e ca n as sume that h has the v alue q − 1 in all its entries and M 1 , . . . , M n/n a are as in Corolla ry 3.1 0. W e als o ass ume that the nonzero column of each M i is the first one. Finally , w e can p ermute the columns of the matrix H = [ B B · · · B ] to obtain the matrix H ′ = [ B 1 B 2 · · · B n/n a ] , 15 where B i has all its columns equal to the i - th column of B . It is stra ightf orward to see that G a nd H ′ are o rthogonal matrices.  Finally , w e summarize the main result of this pa p er . Theorem 3 .12 L et C b e a [ n, k , d ] q c ompletely r e gular c o de with c overing r adius ρ = 1 . L et A b e a gener ator matrix for the r ep et ition [ n a , 1 , n a ] q c o de of length n a and let B b e a p arity che ck matrix of a Hamming c o de with p ar ameters [ n b , k b , 3] q , wher e n = n a n b , n b = ( q m b − 1 ) / ( q − 1) , k b = n b − m b . (i) If d = 1 , then C is the q - re p e ate d c o de of a c ompletely r e gu lar c o de C ′ with c overing r adius ρ ′ = 1 and minimum distanc e d ′ ∈ { 1 , 2 } . (ii) If d = 2 , then n a > 1 and C is e quivalent to a c o de with p arity che ck matrix H = A or H = A ⊕ B . (iii) If d = 3 , then n a = 1 and C is a Hamming c o de and H = B is a p arity che ck matrix for C . (iv) C is a c ompletely t r ansitive c o de. Pro of: W e k now that d ∈ { 1 , 2 , 3 } . W e separa te these three cases: (i) W e hav e prov en this s tatement in Corolla ry 2.4. (ii) This is proven in Prop ositio n 3.11. (iii) O b vious, since C is a p erfect co de. (iv) If d ∈ { 2 , 3 } , by Pr op osition 3.11 and Theor em 3.4, C is eq uiv a lent to a co de C ′ such that Aut( C ′ ) is transitive. Thus, by Lemma 1.3, Aut( C ) is transitive and, by Lemma 1.5, C is completely transitive. If d = 1, then let D be the ‘reduced’ co de, that is, the co de obtained from C b y doing the r everse op eration of the q - r ep eated co de construction. Since the covering r adius of C and D is 1, w e have that C 6 = F n q and D is a completely re g ular co de with d > 1 by Theor e m 2.3. Therefo re D is a completely transitiv e co de. This means that we ca n choos e a set of 16 q n − k − 1 coset leader s of weight one such that they are in the same or bit of Aut( D ). But C has the same num be r of co sets a nd w e ca n choo se the same coset leaders. Since, clear ly , Aut( D ) ⊆ Aut( C ), we hav e tha t these coset leaders are in the same orbit. 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