Constructions of Subsystem Codes over Finite Fields

AL Y , KLAPPENECKER: CO NSTRUCTIONS OF SUBSYSTEM CODES O VER FINITE FIELDS, 2008. 1 Constructions of Subsystem Codes o v er Finite Fields Salah A. Aly and Andrea s Klappenecker Department of Comp uter Science T exas A&M Uni versity , Colle ge Station, TX 77843-31 12, USA Emails: { salah,k lappi } @cs.tamu.ed u Abstract —Subsystem codes protect q uantum infor mation by encoding it in a tensor factor of a sub space of the physical state space. Su bsystem codes generalize all major q uantum error protection schemes, and there for e are especially versatile. Th is paper introduces n umerous c onstructions of subsystem codes. It is sho wn how one can derive su bsystem codes fr om classical cyclic codes. Methods to trade the dimensions of subsystem and co-subystem are in troduced th at maint ain or impro ve the minimum distance. As a consequence, many optimal sub system codes ar e o btained. Furthermore, it is shown ho w gi ven s ubsystem codes can be extended, shortened, or c ombined to yield n ew subsystem codes. These su bsystem code constructions are used to deri ve tables of u pper and lower b ounds on th e su bsystem code parameters. I . I N T RO D U C T I O N Quantum inform ation processing as a growing exciting field has attracted researchers from different disciplines. It utilizes the la ws o f quantum mechan ical ope rations to perform expo - nentially speed y compu tations. In an open system, o ne might wonder how to perform such computations in the presence of decoh erence and noise that disturb quantu m states storing quantum inform ation. Ultimately , the goals of qua ntum error- correcting cod es are to p rotect quan tum states and to allow recovery of qu antum information processed in c omputatio nal operation s of a quan tum computer . Hencefor th, one seeks to design good qu antum co des that can be ef ficien tly utilized for these goals. A well-kn own app roach to derive quantum error-correcting codes from s elf-orth ogon al (or dual-co ntaining) cla ssical codes is called stab ilizer co des, which were introdu ced a decade ago. The stabilizer codes inherit some proper ties of clif f ord gr oup theory , i.e ., they are stab ilized by abelian finite groups. In the seminal paper by Calderbank a t. et [ 7 ], [ 20 ], [ 22 ], various methods of stabilizer code con structions ar e gi ven, along with their pr opagatio n ru les an d tables of uppe r bou nds on their parameters. In a similar tactic, we a lso present subsystem code structures by establishing several metho ds to der i ve them easily from classical codes. Subsystem codes inherit their name fro m the fact that the quan tum code s ar e decomp osed into two systems as explained in Section II . The classes of subsystem codes that we will derive are super ior because they ca n be encod ed and decoded using linear shirt- register operation s. In addition, s ome of these classes turned o ut to be optimal and MDS c odes. Subsystem codes as we prefer to call them were m entioned in th e unp ublished work by Knill [ 14 ], [ 15 ], in which he attempted to generalize the theo ry of qu antum er ror-correcting codes into subsystem co des. Such codes with their stabilizer formalism were reintrod uced recen tly [ 6 ], [ 12 ], [ 16 ], [ 17 ], [ 19 ]. An (( n, K, R, d )) q subsystem code is a K R -dim ensional subspace Q of C q n that is decom posed into a tensor product Q = A ⊗ B of a K -dimension al vector space A an d an R - dimensiona l vector sp ace B such that a ll erro rs of weight less than d can be d etected by A . T he vector spaces A and B are respectively c alled the sub system A and th e co- subsystem B . For some backgroun d on subsystem codes see the next section. This paper is structured as fo llows. In section II , we pre sent a b rief backgroun d o n subsystem code structures an d present the Euclidean and Hermitian con structions. I n section III , we d erive cyclic s ubsystem codes an d provide two generic methods of their constru ctions from classical cyclic codes. Consequently in section IV , we co nstruct families of subsys- tem BCH and RS co des from c lassical BCH an d RS over F q and F q 2 defined using their defining sets. In Sections V , VI , VII , we establish various metho ds of subsystem co de constructio ns by extending and shortening the code lengths and com bining pairs of known cod es, in addition, tables of upper bou nds o n subsystem code parameters are gi ven. Finally , the paper is conclud ed with a d iscussion an d fu ture r esearch d irections in section VIII . Notation. If S is a set, th en | S | denotes th e cardinality of the set S . Let q be a power of a prim e in teger p . W e deno te by F q the finite field with q elem ents. W e use the no tation ( x | y ) = ( x 1 , . . . , x n | y 1 , . . . , y n ) to d enote the concaten ation of two vectors x an d y in F n q . Th e sy mplectic weig ht of ( x | y ) ∈ F 2 n q is defined as swt( x | y ) = { ( x i , y i ) 6 = (0 , 0) | 1 ≤ i ≤ n } . W e d efine swt( X ) = min { swt( x ) | x ∈ X , x 6 = 0 } fo r any nonemp ty sub set X 6 = { 0 } of F 2 n q . The tra ce-symplectic prod uct o f two vectors u = ( a | b ) and v = ( a ′ | b ′ ) in F 2 n q is defin ed as h u | v i s = tr q/p ( a ′ · b − a · b ′ ) , where x · y d enotes the dot produ ct and tr q/p denotes th e trace from F q to the sub field F p . The tr ace-symplectic d ual AL Y , KLAPPENECKER: CONSTRUCTIONS OF SUBSYSTEM CODE S OVER FINITE FIELDS, 2008. 2 of a code C ⊆ F 2 n q is defined as C ⊥ s = { v ∈ F 2 n q | h v | w i s = 0 f or all w ∈ C } . W e define th e Eu clidean inn er pro duct h x | y i = P n i =1 x i y i and the Euclidean dual of C ⊆ F n q as C ⊥ = { x ∈ F n q | h x | y i = 0 for all y ∈ C } . W e also d efine th e H ermitian inner pro duct for vecto rs x, y in F n q 2 as h x | y i h = P n i =1 x q i y i and the Hermitian dual of C ⊆ F n q 2 as C ⊥ h = { x ∈ F n q 2 | h x | y i h = 0 f or all y ∈ C } . I I . B AC K G RO U N D O N S U B S Y S T E M C O D E S In this section we gi ve a quick o verview of subsystem codes. W e a ssume that th e reader is familiar the theo ry of stabilizer codes ov er finite field s, see [ 7 ], [ 11 ], [ 20 ] and the re ferences therein. A. Err ors Let F q denote a finite field with q elemen ts of characteris- tic p . Let {| x i | x ∈ F q } be a fixed ortho norma l b asis of C q with r espect to the standard herm itian inner pro duct, called the computational basis. For a, b ∈ F q , we define the unitary operator s X ( a ) and Z ( b ) o n C q by X ( a ) | x i = | x + a i , Z ( b ) | x i = ω tr( bx ) | x i , where ω = exp(2 π i/p ) is a primitive p th ro ot of u nity and tr is the trace operatio n fro m F q to F p . The set E = { X ( a ) Z ( b ) | a, b ∈ F q } f orms an orthogon al basis o f the operator s acting on C q with respect to the tr ace in ner pro duct, called the error ba sis. The state spa ce of n qua ntum d igits ( or qudits) is gi ven by C q n = C q ⊗ C q ⊗ · · · ⊗ C q . A n err or basis E on C q n is ob tained by tensor ing n op erators in E ; more explicitly , E = { X ( a ) Z ( b ) | a , b ∈ F n q } , where X ( a ) = X ( a 1 ) ⊗ · · · ⊗ X ( a n ) , Z ( b ) = Z ( b 1 ) ⊗ · · · ⊗ Z ( b n ) for a = ( a 1 , . . . , a n ) ∈ F n q and b = ( b 1 , . . . , b n ) ∈ F n q . The set E is not clo sed under m ultiplication, w hence it is not a group . Th e g roup G generated by E is giv en by G = { ω c E = ω c X ( a ) Z ( b ) | a , b ∈ F n q , c ∈ F p } , and G is called the erro r group of C q n . The er ror gro up is an extraspec ial p -grou p. The weight of an erro r in G is given by th e nu mber of non identity tensor comp onents; hence, the weight of ω c X ( a ) Z ( b ) is given by the symplectic weight swt( a | b ) . B. Subsystem Cod es An (( n, K, R, d )) q subsystem code is a subspace Q = A ⊗ B of C q n that is de composed into a ten sor p rodu ct of two vector spaces A an d B of d imension dim A = K and dim B = R such th at all er rors in G of weig ht less than d can be detected by A . W e call A the su bsystem an d B the co-subsystem . The informa tion is exclusi vely encoded in the subsystem A . This yields the attractive feature that error s affecting co -subsystem B a lone ca n be ignored . A particularly fruitful way to construct subsystem codes proceed s by cho osing a norm al sub group N of the err or group G , and this choice deter mines the dim ensions of sub - system and co-sub system as well as the error detection and correction capa bilities of the subsystem code, see [ 12 ]. One can r elate the norm al subgro up N to a classical code, namely N modulo the inter section of N with the center Z ( G ) of G yields the classical c ode X = N / ( N ∩ Z ( G )) . This generalizes the familiar ca se o f stabilizer cod es, where N is an abelian norm al subg roup. It is remarkable that in th e ca se of subsystem codes any classical add itiv e code X can occur . It is most co n venient th at one can also start with any classical additive code and obtain a subsystem code, as is detailed in the fo llowing theo rem fro m [ 12 ]: Theorem 1 . Let C be a c lassical add itive subco de o f F 2 n q such that C 6 = { 0 } and let D deno te its subcod e D = C ∩ C ⊥ s . If x = | C | and y = | D | , then there e xists a sub system co de Q = A ⊗ B such that i) dim A = q n / ( xy ) 1 / 2 , ii) dim B = ( x/y ) 1 / 2 . The minimu m distanc e of subsystem A is given by (a) d = swt(( C + C ⊥ s ) − C ) = swt( D ⊥ s − C ) if D ⊥ s 6 = C ; (b) d = swt( D ⊥ s ) if D ⊥ s = C . Thus, the sub system A ca n detect all err ors in E o f weight less than d , and can corr e ct all err ors in E of weight ≤ ⌊ ( d − 1 ) / 2 ⌋ . Pr oo f: See [ 12 , Theo rem 5]. A subsystem code that is derived with the help of the previous theorem is c alled a Clifford sub system code . W e will assume thr ough out this pap er that all subsystem cod es are Clifford subsystem code s. In particular , this means that the existence of an ( ( n, K , R, d )) q subsystem code imp lies the existence of an additi ve code C ≤ F 2 n q with subcode D = C ∩ C ⊥ s such that | C | = q n R/K , | D | = q n / ( K R ) , and d = swt( D ⊥ s − C ) . A subsystem co de derived from an ad ditive classical code C is called p ure to d ′ if there is no elem ent of sy mplectic weigh t less th an d ′ in C . A subsy stem code is called pu re if it is p ure to the m inimum distance d . W e req uire that an (( n, 1 , R, d )) q subsystem co de m ust be pure. W e also use the bracket notation [[ n, k , r , d ]] q to write the parameters of an (( n, q k , q r , d )) q subsystem code in simpler form. Some authors say that an [[ n, k , r, d ]] q subsystem code has r gauge q udits, but this te rminolog y is slightly confu sing, as th e co-subsystem ty pically do es n ot co rrespond to a state space o f r qudits except per haps in trivial cases. W e will avoid this misleadin g termino logy . An (( n, K, 1 , d )) q subsystem code is also an (( n, K, d )) q stabilizer cod e and vice versa. AL Y , KLAPPENECKER: CONSTRUCTIONS OF SUBSYSTEM CODE S OVER FINITE FIELDS, 2008. 3 Subsystem cod es can be co nstructed fro m the c lassical codes over F q and F q 2 . W e recall the Euclidean and Hermitian construction s fro m [ 3 ], which are easy co nsequen ces of the previous theorem. Lemma 2 (Eu clidean Construction ) . If C is a k ′ -dimensiona l F q -linear code of leng th n that ha s a k ′′ -dimensiona l subcod e D = C ∩ C ⊥ and k ′ + k ′′ < n , th en the r e exists an [[ n, n − ( k ′ + k ′′ ) , k ′ − k ′′ , wt( D ⊥ \ C )]] q subsystem code. Lemma 3 (Hermitian Constru ction) . If C is a k ′ -dimensiona l F q 2 -linear code of length n that has a k ′′ -dimensiona l su bcode D = C ∩ C ⊥ h and k ′ + k ′′ < n , th en the r e exists an [[ n, n − ( k ′ + k ′′ ) , k ′ − k ′′ , wt( D ⊥ h \ C )]] q subsystem code. I I I . C Y C L I C S U B S Y S T E M C O D E S In this section we shall d erive subsystem co des from classical cyclic cod es. W e first recall some d efinitions before embarkin g on the c onstruction o f subsystem c odes. For fur ther details concern ing cyclic codes see fo r instan ce [ 10 ] and [ 18 ]. Let n be a positi ve integer and F q a finite field with q elements such that gcd( n, q ) = 1 . Recall that a linea r code C ⊆ F n q is called cy clic if an d o nly if ( c 0 , . . . , c n − 1 ) in C implies that ( c n − 1 , c 0 , . . . , c n − 2 ) in C . For g ( x ) in F q [ x ] , we write ( g ( x )) to den ote the principal ideal generated by g ( x ) in F q [ x ] . Let π d enote the vector space isomorph ism π : F n q → R n = F q [ x ] / ( x n − 1) gi ven by π (( c 0 , . . . , c n − 1 )) = c 0 + c 1 x + · · · + c n − 1 x n − 1 + ( x n − 1) . A cyclic code C ⊆ F n q is m apped to a princ ipal ideal π ( C ) of the ring R n . For a cyclic code C , the unique monic polynomial g ( x ) in F q [ x ] of the least degree such that ( g ( x )) = π ( C ) is called the generator polyno mial of C . If C ⊆ F n q is a c yclic code w ith gen erator po lynomial g ( x ) , th en dim F q C = n − deg g ( x ) . Since gcd( n, q ) = 1 , there exists a p rimitive n th root of unity α o ver F q ; that is, F q [ α ] is th e splitting field o f the polyno mial x n − 1 over F q . Let u s hencef orth fix this pr imitiv e n th primitive ro ot of unity α . Since the generator poly nomial g ( x ) o f a cyclic c ode C ⊆ F n q is of m inimal degree, it fo llows that g ( x ) divides the polynomial x n − 1 in F q [ x ] . Therefore, the genera tor p olynom ial g ( x ) of a cyclic code C ⊆ F n q can be u niquely specified in terms of a subset T of { 0 , . . . , n − 1 } such tha t g ( x ) = Y t ∈ T ( x − α t ) . The set T is called the d efining set of th e cyclic cod e C (with respect to the primitive n th root of u nity α ). Since g ( x ) is a polyno mial in F q [ x ] , a definin g set is the unio n o f cycloto mic cosets C x , w here C x = { xq i mo d n | i ∈ Z , i ≥ 0 } , 0 ≤ x < n The fo llowing lemma recalls some well-k nown and easily proved facts about defining sets (see e.g. [ 10 ]). Lemma 4. Let C i be a cyclic code o f leng th n over F q with defining set a T i for i = 1 , 2 . Let N = { 0 , 1 , . . . , n − 1 } an d T a 1 = { at mo d n | t ∈ T } for some inte ger a . Then i) C 1 ∩ C 2 has defining set T 1 ∪ T 2 . ii) C 1 + C 2 has de fining set T 1 ∩ T 2 . iii) C 1 ⊆ C 2 if a nd o nly if T 2 ⊆ T 1 . iv) C ⊥ 1 has defining set N \ T − 1 1 . v) C ⊥ h 1 has defining set N \ T − r 1 pr ovided tha t q = r 2 for some positive inte ger r . Notation. Th rough out this section, we d enote by N the set N = { 0 , . . . , n − 1 } . The cyclotomic coset of x will be denoted by C x . If T is a d efining set of a cyclic code of len gth n and a is an in teger , then we den ote hen ceforth by T a the set T a = { at mo d n | t ∈ T } , as in the p revious lemma . W e use a sup erscript, since this no- tation will be frequ ently used in set differences, an d arguably N \ T − q is mo re rea dable than N \ − q T . Now , we shall give a ge neral constru ction fo r subsystem cyclic codes. W e say that a code C is self-ortho gonal if and only if C ⊆ C ⊥ . W e show that if a classical cyclic code is self-ortho gonal, then one can easily construct cyclic subsystem codes. Proposition 5. Let D be a k -d imensional self-orthogonal cyclic code o f len gth n over F q . Let T D and T D ⊥ r espectively denote the definin g sets of D and D ⊥ . If T is a subset of T D \ T D ⊥ that is the un ion of cyclotomic cosets, then one can define a cyc lic co de C o f length n over F q by the d efining set T C = T D \ ( T ∪ T − 1 ) . If r = | T ∪ T − 1 | is in the range 0 ≤ r < n − 2 k , a nd d = min wt( D ⊥ \ C ) , then ther e exists a subsystem code with parameters [[ n, n − 2 k − r , r, d ]] q . Pr oo f: Since D is a self-ortho gonal cyclic code , we have D ⊆ D ⊥ , w hence T D ⊥ ⊆ T D by Le mma 4 iii). Observe that if s is an element of the set S = T D \ T D ⊥ = T D \ ( N \ T − 1 D ) , then − s is an element of S as well. I n particular, T − 1 is a subset of T D \ T D ⊥ . By definitio n, the cyclic cod e C h as the defining set T C = T D \ ( T ∪ T − 1 ) ; thu s, the dual code C ⊥ has the defining set T C ⊥ = N \ T − 1 C = T D ⊥ ∪ ( T ∪ T − 1 ) . Furthermo re, we have T C ∪ T C ⊥ = ( T D \ ( T ∪ T − 1 )) ∪ ( T D ⊥ ∪ T ∪ T − 1 ) = T D ; therefor e, C ∩ C ⊥ = D by Lemma 4 i). Since n − k = | T D | an d r = | T ∪ T − 1 | , we h ave dim F q D = n − | T D | = k and dim F q C = n − | T C | = k + r . Thus, by L emma 2 there exists an F q -linear subsystem co de with parameters [[ n, κ, ρ, d ]] q , w here i) κ = dim D ⊥ − dim C = n − k − ( k + r ) = n − 2 k − r , ii) ρ = dim C − dim D = k + r − k = r , iii) d = min wt( D ⊥ \ C ) , as claim ed. AL Y , KLAPPENECKER : CONSTRUCTIONS OF SUBSYSTE M CODE S OVER FINITE FIELDS, 2008. 4 W e can also derive subsystem codes fro m cyclic co des over F q 2 by using cyclic codes th at are self-ortho gonal with respect to the He rmitian in ner p rodu ct. Proposition 6. Let D be a cyclic co de of le ngth n over F q 2 such that D ⊆ D ⊥ h . Let T D and T D ⊥ h r espectively be the defining set of D a nd D ⊥ h . If T is a subset of T D \ T D ⊥ h that is the u nion of cyclotomic cosets, then on e can define a cyclic cod e C of length n over F q 2 with defin ing set T C = T D \ ( T ∪ T − q ) . If n − k = | T D | a nd r = | T ∪ T − q | with 0 ≤ r < n − 2 k , and d = wt( D ⊥ h \ C ) , th en the r e e xists an [[ n, n − 2 k − r, r , d ]] q subsystem code. Pr oo f: Sinc e D ⊆ D ⊥ h , their defining sets satisfy T D ⊥ h ⊆ T D by Lemm a 4 iii). If s is an elemen t of T D \ T D ⊥ h , then one easily verifies th at − q s (mo d n ) is an element of T D \ T D ⊥ h . Let N = { 0 , 1 , . . . , n − 1 } . Since the cyclic cod e C has the defining set T C = T D \ ( T ∪ T − q ) , its dual code C ⊥ h has the defining set T C ⊥ h = N \ T − q C = T D ⊥ h ∪ ( T ∪ T − q ) . W e notice th at T C ∪ T C ⊥ h = ( T D \ ( T ∪ T − q )) ∪ ( T D ⊥ h ∪ T ∪ T − q ) = T D ; thus, C ∩ C ⊥ h = D by Le mma 4 i). Since n − k = | T D | and r = | T ∪ T − q | , we h ave dim D = n − | T D | = k and dim C = n − | T C | = k + r . T hus, by Lemma 3 ther e exists an [[ n, κ, ρ, d ]] q subsystem co de with i) κ = dim D ⊥ h − dim C = ( n − k ) − ( k + r ) = n − 2 k − r , ii) ρ = dim C − dim D = k + r − k = r , iii) d = min wt( D ⊥ h \ C ) , as claim ed. W e in clude an example to illustrate the construction gi ven in the p revious p ropo sition. Example 7. Con sider th e na rr ow-sense BCH co de D ⊥ h of length n = 31 over F 4 with design ed distanc e 5. The definin g set T D ⊥ h of D ⊥ h is give n by T D ⊥ h = C 1 ∪ C 2 ∪ C 3 ∪ C 4 = C 1 ∪ C 3 , wher e the cyclotomic cosets of 1 and 3 a r e given by C 1 = { 1 , 4 , 16 , 2 , 8 } and C 3 = { 3 , 12 , 17 , 6 , 24 } . If N = { 0 , 1 , . . . , 30 } , then th e defin ing set of th e dua l code D is given b y T D = N \ ( C 15 ∪ C 7 ) = C 0 ∪ C 1 ∪ C 3 ∪ C 5 ∪ C 11 . Ther efor e, D ⊂ D ⊥ h , dim D ⊥ h = 21 a nd dim D = 1 0 . I f we choose T = C 5 , then T − 2 = C 11 , whence the d efining set T C of the code C is given b y T C = T D \ ( C 5 ∪ C 11 ) = C 0 ∪ C 1 ∪ C 3 . It follo ws that dim C = 20 and dim C ⊥ h = 11 . Ther efor e, th e construction o f the pr eviou s p r op osition yield s a BCH subsystem cod e with parameters [[31 , 1 , 10 , ≥ 5 ]] 2 . The gener al prin ciple be hind the previous example yields the following simple r ecipe f or the con struction o f subsystem codes: Choo se a cyclic code (such as a BCH or Reed- Solomon code) with kn own lower b ound δ on the minimu m distanc e that contains its (hermitian) dual code, and use Propo sition 5 (or Proposition 6 ) to derive subsystem co des. T his a pproac h allows one to control the minimum distance d of the subsystem code, since d ≥ δ is guaran teed. Another ad vantage is that one can exploit the cyclic stru cture in encod ing and deco ding algorithm s. T ABLE I S U B S Y S T E M B C H C O D E S T H AT A R E D E R I V E D U S I N G T H E E U C L I D E A N C O N S T R U C T I O N Subsystem Code Paren t BCH Designed Code C distanc e [[15 , 4 , 3 , 3]] 2 [15 , 7 , 5] 2 4 [[15 , 6 , 1 , 3]] 2 [15 , 5 , 7] 2 6 [[31 , 10 , 1 , 5]] 2 [31 , 11 , 11] 2 8 [[31 , 20 , 1 , 3]] 2 [31 , 6 , 15] 2 12 [[63 , 6 , 21 , 7]] 2 [63 , 39 , 9] 2 8 [[63 , 6 , 15 , 7]] 2 [63 , 36 , 11] 2 10 [[63 , 6 , 3 , 7]] 2 [63 , 30 , 13] 2 12 [[63 , 18 , 3 , 7]] 2 [63 , 24 , 15] 2 14 [[63 , 30 , 3 , 5]] 2 [63 , 18 , 21] 2 16 [[63 , 32 , 1 , 5]] 2 [63 , 16 , 23] 2 22 [[63 , 44 , 1 , 3]] 2 [63 , 10 , 27] 2 24 [[63 , 50 , 1 , 3]] 2 [63 , 7 , 31] 2 28 [[15 , 2 , 5 , 3]] 4 [15 , 9 , 5] 4 4 [[15 , 2 , 3 , 3]] 4 [15 , 8 , 6] 4 6 [[15 , 4 , 1 , 3]] 4 [15 , 6 , 7] 4 7 [[15 , 8 , 1 , 3]] 4 [15 , 4 , 10] 4 8 [[31 , 10 , 1 , 5]] 4 [31 , 11 , 11] 4 8 [[31 , 20 , 1 , 3]] 4 [31 , 6 , 15] 4 12 [[63 , 12 , 9 , 7]] 4 [63 , 30 , 15] 4 15 [[63 , 18 , 9 , 7]] 4 [63 , 27 , 21] 4 16 [[63 , 18 , 7 , 7]] 4 [63 , 26 , 22] 4 22 ∗ punctured code + Extended code For examp le, if we start with primiti ve, narrow-sense BCH codes, then Proposition 5 y ields the f ollowing family of subsystem co des: Corollary 8 . Co nsider a primitive, narr ow-sense BCH code of leng th n = q m − 1 with m ≥ 2 o ver F q with d esigned distance δ in the r ange 2 ≤ δ ≤ q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m is odd ] . (1) If T is a sub set of N \  S δ − 1 a =1 ( C a ∪ C − a )  that is a unio n of cyclotomic cosets an d r = | T ∪ T − 1 | with 0 ≤ r < n − 2 k , wher e k = m ⌈ ( δ − 1 )(1 − 1 /q ) ⌉ , the n there exists a n [[ q m − 1 , q m − 1 − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ − r, r, ≥ δ ]] q subsystem cod e. Pr oo f: By [ 5 , Theorem 2], a primitive, narrow-sense BCH code D ⊥ with design ed distance δ in the ra nge ( 1 ) satisfies D ⊆ D ⊥ . By [ 5 , Theorem 7 ], the dim ension of D ⊥ is giv en by dim D ⊥ = q m − 1 − m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ = n − k , whence k = dim D . Let T D and T D ⊥ r espectively d enote the defining sets o f D and D ⊥ . It f ollows from the definition s that T D ⊥ = S δ − 1 a =1 C a and th at T is a subset of N \ ( T D ⊥ ∪ T − 1 D ⊥ ) = ( N \ T − 1 D ⊥ ) \ T D ⊥ = T D \ T D ⊥ . If T C = T D \ ( T ∪ T − 1 ) denotes th e defining set of a c yclic code C , then dim C = k + r . By Prop osition 5 , the re exists an [[ n, n − 2 k − r, r, ≥ δ ]] q subsystem code, which proves the claim. Similarly , we can obtain a herm itian variation of the pre- ceding co rollary with the help of Propo sition 6 . AL Y , KLAPPENECKER : CONSTRUCTIONS OF SUBSYSTE M CODE S OVER FINITE FIELDS, 2008. 5 T ABLE II S U B S Y S T E M B C H C O D E S T H A T A R E D E R I V E D W I T H T H E H E L P O F T H E H E R M I T I A N C O N S T R U C T I O N Subsystem Code Paren t BCH Designed Code C distanc e [[14 , 1 , 3 , 4]] 2 [14 , 8 , 5] 2 2 6 ∗ [[15 , 1 , 2 , 5]] 2 [15 , 8 , 6] 2 2 6 [[15 , 5 , 2 , 3]] 2 [15 , 6 , 7] 2 2 7 [[16 , 5 , 2 , 3]] 2 [16 , 6 , 7] 2 2 7 + [[17 , 8 , 1 , 4]] 2 [17 , 5 , 9] 2 2 4 [[21 , 6 , 3 , 3]] 2 [21 , 9 , 7]] 2 2 6 [[21 , 7 , 2 , 3]] 2 [21 , 8 , 9] 2 2 8 [[31 , 10 , 1 , 5]] 2 [31 , 11 , 11] 2 2 8 [[31 , 20 , 1 , 3]] 2 [31 , 6 , 15] 2 2 12 [[32 , 10 , 1 , 5]] 2 [32 , 11 , 11] 2 2 8 + [[32 , 20 , 1 , 3]] 2 [32 , 6 , 15] 2 2 12 + [[25 , 12 , 3 , 3]] 3 [25 , 8 , 12] 3 2 9 ∗ [[26 , 6 , 2 , 5]] 3 [26 , 11 , 8] 3 2 8 [[26 , 12 , 2 , 4]] 3 [26 , 8 , 13] 3 2 9 [[26 , 13 , 1 , 4]] 3 [26 , 7 , 14] 3 2 14 [[80 , 1 , 17 , 20]] 3 [80 , 48 , 21] 3 2 21 [[80 , 5 , 17 , 17]] 3 [80 , 46 , 22] 3 2 22 ∗ puncture d code + Extended code Corollary 9 . Consider a p rimitive, narr ow-sense BCH code of leng th n = q 2 m − 1 with m 6 = 2 over F q with d esigned distance δ in the r ange 2 ≤ δ ≤ q m − 1 (2) If T is a subset of the set N \  S δ − 1 a =1 ( C a ∪ C − qa )  that is a union of cyclotomic cosets and r = | T ∪ T − q | with 0 ≤ r < n − 2 k , where k = m ⌈ ( δ − 1 )(1 − 1 /q 2 ) ⌉ , then th er e exists a [[ q 2 m − 1 , q 2 m − 1 − 2 m ⌈ ( δ − 1)(1 − 1 /q 2 ) ⌉ − r, r, ≥ δ ]] q subsystem code. Pr oo f: The pr oof is similar to the pro of of the previous corollary , and is a co nsequenc e of [ 5 , Th eorems 4 a nd 7 ] an d Proposition 6 . It is straightfor ward to g eneralize th e previous two corollar- ies to the case of non- primitive BCH code s using the r esults giv en in [ 4 ], [ 2 ]. One of the disadvantages of the cyclic constructio ns is that the p arameter r is restricted to values dictated by the possible cardinalities of th e sets T ∪ T − 1 (or T ∪ T − q ), wher e T is confined to be a unio n of cyclotomic cosets. In the n ext section, we will see how on e can overcome this limitation. W e conclu de this sectio n by giving some examples of the parameters of subsystem BCH codes in T ables I and II . I V . T R A D I N G D I M E N S I O N S O F S U B S Y S T E M A N D C O - S U B S Y S T E M C O D E S In this section we show how one can trad e the dimensions of sub system and co- subsystem to ob tain new codes f rom a giv en subsystem o r stabilizer code. The results ar e obtaine d by exploiting the symplectic geom etry of the space. A re markable consequen ce is that nearly any stabilizer code yie lds a series of sub system cod es. Our first resu lt shows th at one can decrease the dime nsion of the sub system and increase at th e same time the dimensio n of the co-sub system wh ile keeping or increa sing th e minimu m distance o f the subsystem co de. Theorem 10. Let q be a power of a prime p . If th er e exists a n (( n, K , R , d )) q subsystem cod e with K > p that is pu r e to d ′ , then ther e exists a n (( n, K/ p, p R , ≥ d )) q subsystem code that is p ur e to min { d, d ′ } . I f a pure (( n, p, R , d )) q subsystem co de exis ts, then ther e exists a (( n, 1 , p R, d )) q subsystem code. Pr oo f: By definition , an (( n, K , R, d )) q Clif ford subsys- tem code is associated with a classical ad ditiv e co de C ⊆ F 2 n q and its subcod e D = C ∩ C ⊥ s such th at x = | C | , y = | D | , K = q n / ( xy ) 1 / 2 , R = ( x/y ) 1 / 2 , and d = swt( D ⊥ s − C ) if C 6 = D ⊥ s , o therwise d = swt( D ⊥ s ) if D ⊥ s = C . W e h ave q = p m for some positive integer m . Since K and R are po siti ve integers, we h ave x = p s +2 r and y = p s for some integers r ≥ 1 , and s ≥ 0 . There exists an F p -basis of C of the fo rm C = span F p { z 1 , . . . , z s , x s +1 , z s +1 , . . . , x s + r , z s + r } that can be e xtended to a symplectic basis { x 1 , z 1 , . . . , x nm , z nm } of F 2 n q , that is, h x k | x ℓ i s = 0 , h z k | z ℓ i s = 0 , h x k | z ℓ i s = δ k,ℓ for all 1 ≤ k , ℓ ≤ nm , see [ 8 , Th eorem 8. 10.1] . Define an additive code C m = span F p { z 1 , . . . , z s , x s +1 , z s +1 , . . . , x s + r +1 , z s + r +1 } . It f ollows th at C ⊥ s m = span F p { z 1 , . . . , z s , x s + r +2 , z s + r +2 , . . . , x nm , z nm } and D = C m ∩ C ⊥ s m = span F p { z 1 , . . . , z s } . By d efinition, the code C is a subset of C m . The subsystem code defined by C m has th e param eters ( n, K m , R m , d m ) , wh ere K m = q n / ( p s +2 r +2 p s ) 1 / 2 = K /p and R m = ( p s +2 r +2 /p s ) 1 / 2 = pR . For th e claims concer ning minimum d istance an d pu rity , we distinguish two c ases: (a) If C m 6 = D ⊥ s , then K > p and d m = swt( D ⊥ s − C m ) ≥ swt( D ⊥ s − C ) = d . Since by hy pothesis swt( D ⊥ s − C ) = d and swt( C ) ≥ d ′ , and D ⊆ C ⊂ C m ⊆ D ⊥ s by construction , we have swt( C m ) ≥ min { d, d ′ } ; thu s, the subsystem co de is pure to min { d, d ′ } . (b) If C m = D ⊥ s , then K m = 1 = K /p , that is, K = p ; it follows fr om the assumed purity that d = swt( D ⊥ s − C ) = swt( D ⊥ s ) = d m . This proves the claim. For F q -linear su bsystem codes there exists a variation of the p revious theore m which asserts that one can constru ct the resulting subsy stem code such that it is again F q -linear . Theorem 11. Let q be a power of a prime p . If th er e exists a n F q -linear [[ n, k , r, d ]] q subsystem code with k > 1 tha t is pure to d ′ , then there exists an F q -linear [[ n , k − 1 , r + 1 , ≥ d ]] q subsystem co de th at is pu r e to min { d, d ′ } . I f a pure F q -linear [[ n, 1 , r, d ]] q subsystem code exis ts, th en the r e exists an F q - linear [[ n, 0 , r + 1 , d ]] q subsystem code. Pr oo f: The proof is analo gous to the proof of the pr evious theorem, except that F q -bases are used instead of F p -bases. AL Y , KLAPPENECKER : CONSTRUCTIONS OF SUBSYSTE M CODE S OVER FINITE FIELDS, 2008. 6 There exists a pa rtial conv erse of Theo rem 10 , n amely if the subsy stem code is pu re, then it is possible to in crease the dimension of the subsystem and decrease the dimen sion of the co-subsystem while maintaining the same minimum distance. Theorem 12. Let q be a p ower of a prime p . If th er e e xists a pu r e (( n, K , R, d )) q subsystem cod e with R > 1 , then ther e exis ts a pure (( n, pK , R /p, d )) q subsystem cod e. Pr oo f: Suppose that the (( n, K , R , d )) q Clif ford subsys- tem code is associated with a c lassical ad ditiv e code C m = span F p { z 1 , . . . , z s , x s +1 , z s +1 , . . . , x s + r +1 , z s + r +1 } . Let D = C m ∩ C ⊥ s m . W e h ave x = | C m | = p s +2 r +2 , y = | D | = p s , hence K = q n /p r + s and R = p r +1 . Furtherm ore, d = swt( D ⊥ s ) . The co de C = span F p { z 1 , . . . , z s , x s +1 , z s +1 , . . . , x s + r , z s + r } has the subco de D = C ∩ C ⊥ s . Since | C | = | C m | /p 2 , the parameters of the Clif ford sub system cod e associated with C are (( n, pK, R/p , d ′ )) q . Since C ⊂ C m , the minimum distance d ′ satisfies d ′ = swt( D ⊥ s − C ) ≤ swt( D ⊥ s − C m ) = swt( D ⊥ s ) = d. On the other hand , d ′ = swt( D ⊥ s − C ) ≥ s wt( D ⊥ s ) = d , whence d = d ′ . Fu rthermo re, the resulting co de is pure since d = swt( D ⊥ s ) = swt( D ⊥ s − C ) . Replacing F p -bases by F q -bases in the proof of the previous theorem yields the following variation of the previous theorem for F q -linear subsystem codes. Theorem 13. Let q be a po wer of a prime p . If there exists a pur e F q -linear [[ n, k , r , d ]] q subsystem code with r > 0 , then ther e e xists a pur e F q -linear [[ n, k + 1 , r − 1 , d ]] q subsystem code. The purity hypo thesis in Theorem s 12 and 13 is essential, as the next remark sho ws. Remark 14. The Bacon -Shor co de is an impur e [[9 , 1 , 4 , 3]] 2 subsystem code. Ho wever , there do es not e xist any [[9 , 5 , 3]] 2 stabilizer co de. Th us, in general o ne ca nnot omit the purity assumption fr om Th eor ems 12 a nd 1 3 . An [[ n, k , d ]] q stabilizer code can also be regarded as a n [[ n, k , 0 , d ]] q subsystem cod e. W e reco rd this importa nt special case o f the previous th eorems in the next co rollary . Corollary 15. If th er e exists an ( F q -linear) [[ n, k , d ]] q stabi- lizer code that is pur e to d ′ , then there exists for all r in the range 0 ≤ r < k an ( F q -linear) [[ n, k − r, r, ≥ d ]] q subsystem code that is pur e to min { d, d ′ } . I f a pur e ( F q - linear) [[ n, k , r, d ]] q subsystem cod e exists, th en a pure ( F q - linear) [[ n, k + r, d ]] q stabilizer co de exists. W e have shown in [ 4 ], [ 5 ] that a ( primitive or non-pr imitiv e) narrow sense BCH code o f length n over F q contains its d ual code if the de signed distance δ is in the range 2 ≤ δ ≤ δ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m odd ]) . For simplicity , we will proc eed o ur work fo r pr imitiv e nar- row sense BCH co des, however , the gen eralization f or non- primitive BCH codes is a straightfor ward. Corollary 16. If q is power of a prime, m is a positive integer , and 2 ≤ δ ≤ q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m odd ] . Th en ther e exists a subsystem BCH code with p arameters [[ q m − 1 , n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ − r, r , ≥ δ ]] q wher e 0 ≤ r < n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ . Pr oo f: W e know that if 2 ≤ δ ≤ q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m odd ] , then ther e exists a stabilizer co de w ith pa rameters [[ q m − 1 , n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ , ≥ δ ]] q . Let r be an in teger in the ra nge 0 ≤ r < n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ . From [ 1 , Theorem 2], th en there must exist a sub system BCH code with parameters [[ q m − 1 , n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ − r, r , ≥ δ ]] q . W e can also constru ct subsystem BCH codes fr om stabilizer codes u sing the Hermitian con structions. Corollary 1 7. If q is a po wer of a prime, m is a po sitive inte ger , an d δ is an integ er in th e range 2 ≤ δ ≤ δ max = q m +[ m even ] − 1 − ( q 2 − 2)[ m e ven ] , then there exists a subsystem code Q with pa rameters [[ q 2 m − 1 , q 2 m − 1 − 2 m ⌈ ( δ − 1)(1 − 1 /q 2 ) ⌉ − r, r , d Q ≥ δ ]] q that is pure u p to δ , wher e 0 ≤ r < q 2 m − 1 − 2 m ⌈ ( δ − 1 )(1 − 1 /q 2 ) ⌉ . Pr oo f: If 2 ≤ δ ≤ δ max = q m +[ m even ] − 1 − ( q 2 − 2)[ m e ven ] , then exists a classical BCH code with param eters [ q m − 1 , q m − 1 − m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ , ≥ δ ] q which con tains its dual cod e. Fro m [ 1 , Theor em 2 ], the n there must exist a subsystem co de with the given p arameters. V . M D S S U B S Y S T E M C O D E S Recall that an [[ n, k , r , d ]] q subsystem co de derived fro m an F q -linear classical code C ≤ F 2 n q satisfies the Singleto n bound k + r ≤ n − 2 d + 2 , see [ 13 , Theor em 3.6 ]. A subsy stem code attaining the Singleton bound with equ ality is called an MDS su bsystem co de. An important consequence of th e previous theorems is the following simple observation which yields an easy construc- tion of subsystem codes that are optimal amo ng the F q -linear Clif ford sub system co des. Theorem 18 . If th er e exists an F q -linear [[ n, k , d ]] q MDS sta- bilizer cod e, then ther e exists a pure F q -linear [[ n, k − r, r , d ]] q MDS subsystem code for all r in the range 0 ≤ r ≤ k . Pr oo f: An MDS stabilizer code mu st be pure , see [ 20 , Theorem 2] or [ 11 , Corollar y 60]. By Corollary 15 , a p ure F q -linear [[ n, k , d ]] q stabilizer code imp lies the existence of an F q -linear [[ n, k − r, r , d r ≥ d ]] q subsystem cod e that is pur e to d for any r in th e range 0 ≤ r ≤ k . Sin ce the stabilizer cod e is MDS, we h av e k = n − 2 d + 2 . By the Singleto n bou nd, the parameters of the r esulting F q -linear [[ n, n − 2 d + 2 − r , r, d r ]] q subsystem codes must satisfy ( n − 2 d + 2 − r ) + r ≤ n − 2 d r + 2 , which shows that the minim um distance d r = d , as claimed . AL Y , KLAPPENECKER : CONSTRUCTIONS OF SUBSYSTE M CODE S OVER FINITE FIELDS, 2008. 7 Remark 1 9. W e conjectu r e that F q -linear MDS sub system codes ar e actually optimal amo ng a ll sub system codes, but a pr o of that the Sin gleton bo und ho lds for general sub system codes remains elusive. In the next corollar y , we g iv e a few examp les of MD S su b- system codes that can be obtained from Theor em 18 . These are the first families of M DS subsystem co des (thou gh spora dic examples of M DS subsy stem codes h ave been established before, see e.g. [ 3 ], [ 6 ]) . Corollary 20. i) An F q -linear pure [[ n, n − 2 d + 2 − r, r , d ]] q MDS subsystem cod e e xists for all n , d , a nd r such that 3 ≤ n ≤ q , 1 ≤ d ≤ n/ 2 + 1 , and 0 ≤ r ≤ n − 2 d + 1 . ii) An F q -linear pur e [[( ν +1) q , ( ν + 1) q − 2 ν − 2 − r , r, ν +2 ]] q MDS sub system code exists for all ν and r such th at 0 ≤ ν ≤ q − 2 and 0 ≤ r ≤ ( ν + 1) q − 2 ν − 3 . iii) An F q -linear pure [[ q − 1 , q − 1 − 2 δ − r , r, δ + 1]] q MDS subsystem code e xists for all δ a nd r such that 0 ≤ δ < ( q − 1 ) / 2 a nd 0 ≤ r ≤ q − 2 δ − 1 . iv) An F q -linear pure [[ q, q − 2 δ − 2 − r ′ , r ′ , δ + 2]] q MDS subsystem co de exists for all 0 ≤ δ < ( q − 1) / 2 and 0 ≤ r ′ < q − 2 δ − 2 . v) An F q -linear pure [[ q 2 − 1 , q 2 − 2 δ − 1 − r , r, δ + 1]] q MDS subsystem code exists for all δ and r in the range 0 ≤ δ < q − 1 an d 0 ≤ r < q 2 − 2 δ − 1 . vi) An F q -linear p ur e [[ q 2 , q 2 − 2 δ − 2 − r ′ , r ′ , δ + 2]] q MDS subsystem cod e exists for all δ a nd r ′ in the range 0 ≤ δ < q − 1 and 0 ≤ r ′ < q 2 − 2 δ − 2 . Pr oo f: i) By [ 9 , Theo rem 14], th ere exist F q -linear [[ n, n − 2 d + 2 , d ]] q stabilizer cod es for a ll n and d such that 3 ≤ n ≤ q an d 1 ≤ d ≤ n/ 2 + 1 . Th e claim fo llows fr om Theorem 18 . ii) By [ 21 , Theore m 5], ther e exist a [[( ν + 1) q , ( ν + 1) q − 2 ν − 2 , ν + 2]] q stabilizer cod e. I n this case, th e code is derived fr om an F q 2 -linear co de X of leng th n over F q 2 such tha t X ⊆ X ⊥ h . T he claim follows fro m Lem ma 29 an d Theorem 18 . iii) ,iv) Th ere exist F q -linear stabilizer codes with p arameters [[ q − 1 , q − 2 δ − 1 , δ + 1]] q and [[ q , q − 2 δ − 2 , δ + 2]] q for 0 ≤ δ < ( q − 1) / 2 , see [ 9 , Theorem 9]. Theorem 18 yields the claim. v) ,vi) The re exist F q -linear stabilizer co des with parame ters [[ q 2 − 1 , q 2 − 2 δ − 1 , δ + 1]] q and [[ q 2 , q 2 − 2 δ − 2 , δ + 2]] q . for 0 ≤ δ < q − 1 b y [ 9 , Theor em 10]. The claim fo llows from Theorem 18 . The existence of the co des in i) are merely established by a non-co nstructive Gilbe rt-V arshamov type coun ting argume nt. Howe ver , th e result is interesting, as it asserts that th ere exist for example [[6 , 1 , 1 , 3]] q subsystem codes for all prime powers q ≥ 7 , [[7 , 1 , 2 , 3 ]] q subsystem cod es fo r all p rime p owers q ≥ 7 , and o ther short subsystem co des that one sh ould compar e with a [[5 , 1 , 3]] q stabilizer code. If the syndro me calculation is simpler, then su ch subsystem cod es co uld b e of practical value. The subsystem co des given in ii)-vi) o f the p revious coro l- lary are co nstructively established. The subsystem codes in ii) a re der iv ed fr om Reed-Mu ller codes, and in iii)-v i) fro m T ABLE III O P T I M A L P U R E S U B S Y S T E M C O D E S Subsystem Codes Paren t Code (RS Code) [[8 , 1 , 5 , 2]] 3 [8 , 6 , 3] 3 2 [[8 , 4 , 2 , 2]] 3 [8 , 3 , 6] 3 2 [[8 , 5 , 1 , 2]] 3 [8 , 2 , 7] 3 2 [[9 , 1 , 4 , 3]] 3 [9 , 6 , 4] † 3 2 , δ = 3 [[9 , 4 , 1 , 3]] 3 [9 , 3 , 7] † 3 2 , δ = 6 [[15 , 1 , 10 , 3]] 4 [15 , 12 , 4] 4 2 [[15 , 9 , 2 , 3]] 4 [15 , 4 , 12] 4 2 [[15 , 10 , 1 , 3]] 4 [15 , 3 , 13] 4 2 [[16 , 1 , 9 , 4]] 4 [16 , 12 , 5] † 4 2 , δ = 4 [[24 , 1 , 17 , 4]] 5 [24 , 20 , 5] 5 2 [[24 , 16 , 2 , 4]] 5 [24 , 5 , 20] 5 2 [[24 , 17 , 1 , 4]] 5 [24 , 4 , 21] 5 2 [[24 , 19 , 1 , 3]] 5 [24 , 3 , 22] 5 2 [[24 , 21 , 1 , 2]] 5 [24 , 2 , 23] 5 2 [[23 , 1 , 18 , 3]] 5 [23 , 20 , 4] ∗ 5 2 , δ = 5 [[23 , 16 , 3 , 3]] 5 [23 , 5 , 19] ∗ 5 2 , δ = 20 [[48 , 1 , 37 , 6]] 7 [48 , 42 , 7] 7 2 * Puncture d code † Extended code Reed-Solomo n codes. There exists an overlap between the parameters g iv en in ii) a nd in iv), but we list her e both, since each co de co nstruction has its own merits. Remark 21. B y Theorem 13 , p ur e MDS sub system codes ca n always be derived fr o m MDS stabilizer codes, see T able II I . Ther efor e, one can derive in fact a ll possible parameter sets of pu r e MDS subsystem codes with the help of Th eor em 18 . Remark 22 . I n the case of stabilizer code s, a ll MDS codes must b e p ur e. F or su bsystem co des this is not true, as the [[9 , 1 , 4 , 3]] 2 subsystem cod e shows. F indin g such impure F q - linear [[ n, k , r, d ]] q MDS subsystem co des with k + r = n − 2 d + 2 is a particularly inter esting challenge. Recall that a pur e subsy stem code is called perfec t if and on ly if it attain s th e Hamm ing bo und with equality . W e co nclude this section with the following con sequence o f Theorem 18 : Corollary 23. If ther e exists an F q -linear pur e [[ n, k , d ]] q stabilizer code tha t is perfect, th en ther e exist s a pur e F q - linear [[ n, k − r , r, d ]] q perfect sub system code for a ll r in the range 0 ≤ r ≤ k . V I . E X T E N D I N G A N D S H O RT E N I N G S U B S Y S T E M C O D E S In Sectio n IV , we showed how on e can derive new subsy s- tem cod es fro m known ones by m odifyin g the dime nsion of the subsystem an d co-subsy stem. In this sectio n, we deri ve new subsystem co des f rom known on es by extending and shortening the length of the cod e. Theorem 24. If there e xists a n (( n, K, R, d )) q Cliffor d sub- system code with K > 1 , th en there exists an (( n + 1 , K , R, ≥ d )) q subsystem code that is pur e to 1. Pr oo f: W e first n ote that fo r any additiv e subcode X ≤ F 2 n q , w e ca n defin e an additi ve code X ′ ≤ F 2 n +2 q by X ′ = { ( aα | b 0) | ( a | b ) ∈ X , α ∈ F q } . AL Y , KLAPPENECKER : CONSTRUCTIONS OF SUBSYSTE M CODE S OVER FINITE FIELDS, 2008. 8 W e hav e | X ′ | = q | X | . Further more, if ( c | d ) ∈ X ⊥ s , th en ( cα | d 0) is contain ed in ( X ′ ) ⊥ s for all α in F q , whence ( X ⊥ s ) ′ ⊆ ( X ′ ) ⊥ s . By comp aring cardinalities we find that equality must h old; in other words, we have ( X ⊥ s ) ′ = ( X ′ ) ⊥ s . By Th eorem 1 , there are two add iti ve cod es C an d D associated with an (( n, K, R, d )) q Clif ford subsystem cod e such tha t | C | = q n R/K and | D | = | C ∩ C ⊥ s | = q n / ( K R ) . W e can derive from the co de C two new additive co des of length 2 n + 2 over F q , nam ely C ′ and D ′ = C ′ ∩ ( C ′ ) ⊥ s . Th e codes C ′ and D ′ determine a (( n + 1 , K ′ , R ′ , d ′ )) q Clif ford subsystem co de. Sin ce D ′ = C ′ ∩ ( C ′ ) ⊥ s = C ′ ∩ ( C ⊥ s ) ′ = ( C ∩ C ⊥ s ) ′ , we have | D ′ | = q | D | . Furtherm ore, we have | C ′ | = q | C | . It follows f rom The orem 1 that (i) K ′ = q n +1 / p | C ′ || D ′ | = q n / p | C || D | = K , (ii) R ′ = ( | C ′ | / | D ′ | ) 1 / 2 = ( | C | / | D | ) 1 / 2 = R , (iii) d ′ = swt(( D ′ ) ⊥ s \ C ′ ) ≥ swt(( D ⊥ s \ C ) ′ ) = d . Since C ′ contains a vector ( 0 α | 0 0 ) of weig ht 1 , the resulting subsystem co de is pure to 1. Corollary 25. If ther e e xists an [[ n, k , r , d ]] q subsystem code with k > 0 an d 0 ≤ r < k , then there exists a n [[ n + 1 , k , r, ≥ d ]] q subsystem code that is pur e to 1. W e can also shorten the leng th of a subsystem code in a simple way as shown in the following Theore m. Theorem 26 . If a pure (( n, K, R, d )) q subsystem cod e exists, then there exists a pur e (( n − 1 , q K , R , d − 1)) q subsystem code. Pr oo f: By [ 3 , Lemma 10], the existence of a p ure Clifford subsystem code with par ameters (( n, K, R, d )) q implies the existence o f a pu re (( n, K R, d )) q stabilizer co de. It follows from [ 11 , Lem ma 70] that there exist a pu re (( n − 1 , q K R, d − 1)) q stabilizer co de, which can be r egarded as a p ure (( n − 1 , q K R, 1 , d − 1)) q subsystem code. Thus, there exists a pur e (( n − 1 , q K, R, d − 1 )) q subsystem code by Theo rem 12 , which proves th e claim. In bracket notation , th e previous theorem states that the existence of a p ure [[ n, k , r, d ]] q subsystem c ode implies the existence of a pure [[ n − 1 , k + 1 , r, d − 1 ]] q subsystem co de. V I I . C O M B I N I N G S U B S Y S T E M C O D E S In this section, we show h ow one c an o btain a new subsys- tem code b y comb ining two given subsystem cod es in various ways. Theorem 27. If ther e e xists a pur e [[ n 1 , k 1 , r 1 , d 1 ]] 2 subsystem code and a pur e [[ n 2 , k 2 , r 2 , d 2 ]] 2 subsystem code such tha t k 2 + r 2 ≤ n 1 , then ther e e xist subsystem codes wi th parameters [[ n 1 + n 2 − k 2 − r 2 , k 1 + r 1 − r, r , d ]] 2 for all r in the range 0 ≤ r < k 1 + r 1 , where the minimum distance d ≥ min { d 1 , d 1 + d 2 − k 2 − r 2 } . Pr oo f: Since there exist pure [[ n 1 , k 1 , r 1 , d 1 ]] 2 and [[ n 2 , k 2 , r 2 , d 2 ]] 2 subsystem codes with k 2 + r 2 ≤ n 1 , it f ollows from Theorem 12 that there exist stabilizer codes with the parameters [[ n 1 , k 1 + r 1 , d 1 ]] 2 and [[ n 2 , k 2 + r 2 , d 2 ]] 2 such that k 2 + r 2 ≤ n 1 . The refore, there exists an [[ n 1 + n 2 − k 2 − r 2 , k 1 + r 1 , d ]] 2 stabilizer cod e with minimum distance d ≥ min { d 1 , d 1 + d 2 − k 2 − r 2 } by [ 7 , Theorem 8]. It follows from Theorem 1 0 that there exists [[ n 1 + n 2 − k 2 − r 2 , k 1 + r 1 − r, r , ≥ d ]] 2 subsystem codes f or all r in the range 0 ≤ r < k 1 + r 1 . Theorem 2 8. Let Q 1 and Q 2 be two pur e subsystem codes with p arameters [[ n , k 1 , r 1 , d 1 ]] q and [[ n, k 2 , r 2 , d 2 ]] q , r espec- tively . If Q 2 ⊆ Q 1 , then th er e exists pur e subsystem co des with parameters [[2 n, k 1 + k 2 + r 1 + r 2 − r, r , d ]] q for all r in the range 0 ≤ r ≤ k 1 + k 2 + r 1 + r 2 , wher e the minimum d istance d ≥ min { d 1 , 2 d 2 } . Pr oo f: By assum ption, there exists a pure [[ n, k i , r i , d i ]] q subsystem code, which implies the existence of a pure [[ n, k i + r i , d i ]] q stabilizer code by Theorem 12 , where i ∈ { 1 , 2 } . By [ 11 , Lemm a 74], there exists a pure stabilizer code with parameter s [[2 n, k 1 + k 2 + r 1 + r 2 , d ]] q such tha t d ≥ min { 2 d 2 , d 1 } . By Theor em 10 , there exist a pu re subsystem code with p arameters [[2 n, k 1 + k 2 + r 1 + r 2 − r, r , d ]] q for all r in the range 0 ≤ r ≤ k 1 + k 2 + r 1 + r 2 , which proves the claim. Further analy sis of pro pagation rules of subsystem code construction s, tables of upper an d lo we r bounds, an d s hort subsystem co des are presented in [ 2 ]. V I I I . C O N C L U S I O N A N D D I S C U S S I O N Subsystem co des ar e amon g the most versatile too ls in quantum error-correction , since they allo w one to combin e the passi ve error-correction fo und in deco herenc e f ree s ub- spaces and noiseless subsy stems with the acti ve err or-control methods of quantum error-correcting code s. In this paper w e demonstra te several meth ods of subsystem co de constru ctions over b inary and non binary fields. The subclass of Clifford subsystem codes that was studied in this pap er is o f particular interest becau se o f the close conn ection to classical err or- correcting codes. As Theorem 1 shows, one can derive f rom each ad ditive code o ver F q an Clif f ord subsystem code. Th is offers mor e fle xibility than the sl ightly rigid framework of stabilizer cod es. W e showed that any F q -linear MDS stabilizer co de yields a series of pu re F q -linear MDS subsystem co des. These AL Y , KLAPPENECKER : CONSTRUCTIONS OF SUBSYSTE M CODE S OVER FINITE FIELDS, 2008. 9 codes are kno wn to be o ptimal among the F q -linear Clif ford subsystem codes. W e conjecture that the Singleton boun d holds in ge neral fo r s ubsystem codes. T here is quite some evidence for this fact, as pur e Clifford subsystem co des and F q -linear Clifford su bsystem codes a re known to obey this bound . W e have establishe d a num ber o f sub system code construction s. I n particular , we ha ve s hown ho w one can deri ve subsystem codes from stabilizer code s. I n combin ation with the propagation rules that we h av e d erived, o ne can easily create table s with the be st known subsystem codes. Further propag ation rules and examp les of such tables are g iv en in [ 2 ], and will appear in an expa nded version of this paper . I X . AC K N OW L E D G M E N T S This research was su pported by NSF grant CCF-062220 1 and NSF CAREER award CCF-034 7310. Part o f this p aper is appeared in Proceedings of 2008 IEEE International Sym- posium o n In formatio n Theory , ISIT’ 08, T oronto, CA, July 2008. R E F E R E N C E S [1] S. A. Aly and A. Klappene cker . Subsystem code construct ions. In Proc. 2008 IE EE International Symposium on Informatio n Theory , T or onto, Canada , pages 369–373, July 2008. [2] S. A. Aly . Quantum error control codes. Ph.D Dissertation, T ex as A&M Univer sity , January 2008. [3] S. A. Aly , A. Klapp eneck er , and P . K. Sarve palli. Subsystem codes. 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A P P E N D I X W e recall that the Hermitian construction of sta bilizer codes yields F q -linear stabilizer c odes, a s can be seen from the following reformulation of [ 9 , Corollary 2]. Lemma 29 ([ 9 ]) . If there exists an F q 2 -linear code X ⊆ F n q 2 such that X ⊆ X ⊥ h , then there exists an F q -linear c ode C ⊆ F 2 n q such that C ⊆ C ⊥ s , | C | = | X | , swt( C ⊥ s − C ) = wt( X ⊥ h − X ) and swt( C ) = wt( X ) . Pr oof: Let { 1 , β } be a basis of F q 2 / F q . Then tr q 2 /q ( β ) = β + β q is an element β 0 of F q ; hence, β q = − β + β 0 . Let C = { ( u | v ) | u, v ∈ F n q , u + β v ∈ X } . It follows from this definition that | X | = | C | and that wt( X ) = swt( C ) . Furthermore, if u + β v and u ′ + β v ′ are elements of X with u, v, u ′ , v ′ in F n q , then 0 = ( u + β v ) q · ( u ′ + β v ′ ) = u · u ′ + β q +1 v · v ′ + β 0 v · u ′ + β ( u · v ′ − v · u ′ ) . On the right hand side, all terms but the last are in F q ; henc e we must have ( u · v ′ − v · u ′ ) = 0 , which shows that ( u | v ) ⊥ s ( u ′ | v ′ ) , whence C ⊆ C ⊥ s . Expanding X ⊥ h in the basis { 1 β } yields a code C ′ ⊆ C ⊥ s , and we must have equality by a dimension argument. Since the basis expansion is isometric, it follows that swt( C ⊥ s − C ) = wt( X ⊥ h − X ) . The F q -linearity of C is a direct consequence of the de finition of C .