Propagation Rules of Subsystem Codes

Propagation Rules of S ubsys tem Codes Salah A. Aly Departmen t of Co mputer Science T exas A&M Univ ersity College Station, TX 77843 , USA Email: salah @cs.tamu.edu Abstract —W e demonstrate pro pagation rules o f subsystem code constru ctions by extending, shortenin g and combining given subsystem codes. Giv en an [[ n, k, r, d ]] q subsystem code, we drive new subsystem codes with parameters [[ n + 1 , k , r, ≥ d ]] q , [[ n − 1 , k + 1 , r, ≥ d − 1]] q , [[ n, k − 1 , r + 1 , d ]] q . W e present the sh ort subsystem codes. The interested readers shall consult our co mpanion papers for up per and lower bounds on subsystem codes parameters, and intro duction, trading di mensions, families, and refer ences on subsystem codes [1], [2], [3] and references therein. Subsystem Codes. Let H b e the Hilber t space C q n = C q ⊗ C q ⊗ ... ⊗ C q . Let Q b e a q uantum cod e such th at H = Q ⊕ Q ⊥ , where Q ⊥ is the o rthogo nal c omplemen t of Q . Recall definition of th e error mo del acting in qu bits [4], [3]. W e can define the subsystem code Q as f ollows Definition 1: An [[ n, k , r , d ]] q subsystem code is a d ecom- position o f the subspace Q into a ten sor product o f two vector spaces A an d B such tha t Q = A ⊗ B , wh ere dim A = k an d dim B = r . T he code Q is able to detect all er rors of weigh t less than d on subsystem A . Subsystem c odes can be constructed from th e classical codes over F q and F q 2 . The Euclidean construction of sub- system code is given as f ollows [1], [3]. Lemma 2 (E uclidean Constructio n): If C is a k ′ - dimensiona l F q -linear code of length n that ha s a k ′′ - dimensiona l sub code D = C ∩ C ⊥ and k ′ + k ′′ < n , then there exists an [[ n, n − ( k ′ + k ′′ ) , k ′ − k ′′ , wt( D ⊥ \ C )]] q subsystem code. I . S U B S Y S T E M C O D E S V E R S . C O - S U B S Y S T E M C O D E S In th is section we show how one can trade the dim ensions of subsystem and co-subsystem to obtain new codes from a giv en subsystem or stabilizer code. The results a re obtaine d by exploiting the symplectic geom etry of the space. A rem arkable consequen ce is that near ly any stabilizer cod e yields a series of su bsystem c odes. Our first result shows that on e can d ecrease the dimension of the sub system and incr ease at the same time the dimension of the c o-subsystem wh ile keeping or incr easing th e minimu m distance of the subsystem code. Theorem 3: Let q be a po wer of a prim e p . If there exists an (( n, K, R , d )) q subsystem code with K > p that is pure to d ′ , then there exists an (( n, K /p, p R , ≥ d )) q subsystem cod e tha t is pur e to m in { d, d ′ } . If a pure ( ( n, p, R, d )) q subsystem cod e exists, then there exists a (( n, 1 , pR, d )) q subsystem code. Pr o of: See [1], [2] Replacing F p -bases by F q -bases in the pr oof of the pre vious theorem yield s the following variation of the previous theorem for F q -linear subsystem co des. Theorem 4: Let q be a p ower of a prime p . If ther e exists a pure F q -linear [[ n, k , r, d ]] q subsystem code with r > 0 , then there exists a p ure F q -linear [[ n, k + 1 , r − 1 , d ]] q subsystem code. Pr o of: See [1], [2] Theorem 5 (Gene ric meth os): If th ere exists an ( F q -linear) [[ n, k , d ]] q stabilizer code that is pure 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 th at is pure to min { d, d ′ } . If a pure ( F q - linear) [[ n, k , r, d ]] q subsystem c ode exists, th en a pure ( F q - linear) [[ n, k + r , d ]] q stabilizer co de exists. Pr o of: See [1], [2] Using th is theo rem we can derive many families o f sub system codes deriv ed f rom families o f stabilizer codes as sho wn in T able 1 I I . P RO PAG AT I O N R U L E S Let C 1 ≤ F n q and C 2 F n q be two classical co des de fined over F q . The dir ect sum of C 1 and C 2 is a code C ≤ F 2 n q defined as f ollows C = C 1 ⊕ C 2 = { uv | u ∈ C 1 , v ∈ C 2 } . (1) In a matr ix fo rm the cod e C can be described as C =  C 1 0 0 C 2  An [ n, k 1 , d 1 ] q classical co de C 1 is a subcod e in an [ c, k 2 , d 2 ] q if every cod ew ord v in C 1 is also a codew ord in C 2 , hence k 1 ≤ k 2 . W e say that an [[ n, k 1 , r 1 , d 1 ]] q subsystem code Q 1 is a sub code in an [[ n, k 2 , r 2 , d 2 ]] q subsystem cod e Q 2 if e very co dew ord | v i in Q 1 is also a codeword in Q 2 and k 1 + r 1 ≤ k 2 + r 1 . Notation. Let q be a power o f a prim e integer p . W e denote by F q the finite field with q elements. W e use the no tation ( x | y ) = ( x 1 , . . . , x n | y 1 , . . . , y n ) to deno te the concatenatio n of two vectors x and y in F n q . The symp lectic weight of ( x | y ) ∈ F 2 n q is defined as swt( x | y ) = { ( x i , y i ) 6 = (0 , 0) | 1 ≤ i ≤ n } . 2 Family Stabilizer [[ n , k , d ]] q Subsystem [[ n, k − r , r, d ]] q , k > r ≥ 0 Short MDS [[ n, n − 2 d + 2 , d ]] q [[ n, n − 2 d + 2 − r, r , d ]] q Hermitian Ham ming [[ n, n − 2 m, 3]] q m ≥ 2 , [[ n, n − 2 m − r, r , 3]] q Euclidean Hamming [[ n , n − 2 m, 3]] q [[ n, n − 2 m − r, r , 3]] q Melas [[ n, n − 2 m, ≥ 3]] q [[ n, n − 2 m − r, r , ≥ 3]] q Euclidean BCH [[ n, n − 2 m ⌈ ( δ − 1 )(1 − 1 / q ) ⌉ , ≥ δ ]] q [[ n, n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ − r, r , ≥ δ ]] q Hermitian BCH [[ n, n − 2 m ⌈ ( δ − 1)(1 − 1 /q 2 ) ⌉ , ≥ δ ]] q [[ n, n − 2 m ⌈ ( δ − 1 )(1 − 1 / q 2 ) ⌉ − r , r , ≥ δ ]] q Punctured MDS [[ q 2 − q α, q 2 − q α − 2 ν − 2 , ν + 2 ]] q [[ q 2 − q α, q 2 − q α − 2 ν − 2 − r, r, ν + 2]] q Euclidean MDS [[ n, n − 2 d + 2]] q [[ n, n − 2 d + 2 − r, r ]] q Hermitian MDS [[ q 2 − s , q 2 − s − 2 d + 2 , d ]] q [[ q 2 − s , q 2 − s − 2 d + 2 − r, r , d ]] q T wisted [[ q r , q r − r − 2 , 3]] q [[ q r , q r − r − 2 − r, r , 3]] q Extended twisted [[ q 2 + 1 , q 2 − 3 , 3]] q [[ q 2 + 1 , q 2 − 3 − r, r , 3]] q Perfect [[ n, n − s − 2 , 3]] q [[ n, n − s − 2 − r, r , 3]] q [[ n, n − s − 2 , 3]] q [[ n, n − s − 2 − r, r , 3]] q Fig. 1. Famili es o f subsystem codes from st abiliz er codes W e define swt( X ) = min { swt( x ) | x ∈ X , x 6 = 0 } for a ny nonemp ty subset X 6 = { 0 } of F 2 n q . The tr ace-symplec tic product o f two vecto rs u = ( a | b ) and v = ( a ′ | b ′ ) in F 2 n q is d efined as h u | v i s = tr q/p ( a ′ · b − a · b ′ ) , where x · y denote s the dot prod uct and tr q/p denotes the trace fr om F q to the subfield F p . The trace-sym plectic d ual of a cod e C ⊆ F 2 n q is defined as C ⊥ s = { v ∈ F 2 n q | h v | w i s = 0 for 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 d ual o f C ⊆ F n q as C ⊥ = { x ∈ F n q | h x | y i = 0 fo r all y ∈ C } . W e also define the Hermitian inner produ ct for vectors x, y in F n q 2 as h x | y i h = P n i =1 x q i y i and the Herm itian d ual of C ⊆ F n q 2 as C ⊥ h = { x ∈ F n q 2 | h x | y i h = 0 for all y ∈ C } . Theorem 6: Let C be a classical additive subco de o f F 2 n q such that C 6 = { 0 } and let D denote its s ubcod e D = C ∩ C ⊥ s . If x = | C | and y = | D | , then ther e exists a subsy stem co de Q = A ⊗ B such that i) dim A = q n / ( xy ) 1 / 2 , ii) dim B = ( x/y ) 1 / 2 . The m inimum d istance o f sub system A is given by (a) d = swt(( C + C ⊥ s ) − C ) = swt( D ⊥ s − C ) if D ⊥ s 6 = C ; (b) d = s wt( D ⊥ s ) if D ⊥ s = C . Thus, the subsy stem A can detect all errors in E of weight less than d , and can corr ect all errors in E of weight ≤ ⌊ ( d − 1 ) / 2 ⌋ . A. Extending Subsystem Cod es W e derive new subsystem code s fro m known ones by extending and shortenin g the length o f th e code. Theorem 7: If there exists an (( n, K, R, d )) q Clif ford sub- system code with K > 1 , then there exists an (( n + 1 , K, R, ≥ d )) q subsystem code that is pure to 1 . Pr o of: W e first no te that for any add iti ve subco de X ≤ F 2 n q , we ca n d efine an additive code X ′ ≤ F 2 n +2 q by X ′ = { ( aα | b 0) | ( a | b ) ∈ X , α ∈ F q } . W e have | X ′ | = q | X | . Fu rthermor e, if ( c | e ) ∈ X ⊥ s , then ( cα | e 0) is contained in ( X ′ ) ⊥ s for all α in F q , when ce ( X ⊥ s ) ′ ⊆ ( X ′ ) ⊥ s . By comparin g cardinalities we find that equality m ust hold; in other words, we have ( X ⊥ s ) ′ = ( X ′ ) ⊥ s . By Theorem 6, there are two additi ve codes C and D associated with an (( n, K, R , d )) q Clif ford sub system cod e such th at | 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 cod es of length 2 n + 2 ov er F q , na mely C ′ and D ′ = C ′ ∩ ( C ′ ) ⊥ s . Th e codes C ′ and D ′ determine a (( n + 1 , K ′ , R ′ , d ′ )) q Clif ford subsystem code. Since D ′ = C ′ ∩ ( C ′ ) ⊥ s = C ′ ∩ ( C ⊥ s ) ′ = ( C ∩ C ⊥ s ) ′ , we have | D ′ | = q | D | . Furthermo re, we have | C ′ | = q | C | . It follows from T heorem 6 th at (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 weigh t 1 , the resulting subsystem code is pure to 1 . Corollary 8 : If there exists an [[ n, k , r, d ]] q subsystem code with k > 0 and 0 ≤ r < k , then there exists an [[ n + 1 , k , r , ≥ d ]] q subsystem code th at is pu re to 1. 3 B. Shortening Subsystem Codes W e can also shorten the leng th of a subsystem code and still tra de the dimension s of the new subsystem c ode and its co-subsystem code as shown in the following Lemma. Theorem 9: If an (( n, K , R , d )) q pure subsystem code Q exists, then there is a pu re subsystem cod e Q p with param eters (( n − 1 , q K , R, ≥ d − 1)) q . Pr o of: W e k now that existence of the p ure sub system code Q with param eters (( n, K , R, d )) q implies existence of a pure stabilizer code with p arameters (( n, K R, ≥ d )) q for n ≥ 2 and d ≥ 2 from [2, Theorem 2.]. By [5, Theorem 70], there exist a pur e stabilizer code with par ameters (( n − 1 , q K R, ≥ d − 1 )) q . This stabilizer cod e can be seen as (( n − 1 , q K R, 0 , ≥ d − 1)) q subsystem c ode. By using [2, Theorem 2.], there exists a pure F q -linear subsystem code with param eters (( n − 1 , q K , R , ≥ d − 1)) q that p roves the c laim. Analog o f the pr evious Theorem is the following Lemm a. Lemma 1 0: I f an F q -linear [[ n, k , r, d ]] q pure subsystem code Q exists, then there is a pure subsy stem code Q p with parameters [[ n − 1 , k + 1 , r, ≥ d − 1]] q . Pr o of: W e k now that existence of the p ure sub system code Q implies existence of a pure stabilizer code with parameters [[ n, k + r , ≥ d ]] q for n ≥ 2 and d ≥ 2 by using [2, Theor em 2. an d Theo rem 5.]. By [5, Theorem 7 0], there exist a pure stabilizer cod e with parameter s [[ n − 1 , k + r + 1 , ≥ d − 1]] q . T his stab ilizer cod e can be seen as an [[ n − 1 , k + r + 1 , 0 , ≥ d − 1]] q subsystem co de. By using [2, Theorem 3.], there e xists a pure F q -linear subsystem code with parameters [[ n − 1 , k + 1 , r , ≥ d − 1]] q that pr oves the claim. W e can also prove the pre vious Theo rem by defining a new code C p from th e code C as follows. Theorem 11: If the re exists a pur e subsystem code Q = A ⊗ B with parameters (( n, K, R, d )) q with n ≥ 2 and d ≥ 2 , then there is a subsystem code Q p with parameter s (( n − 1 , K , q R , ≥ d − 1)) q . Pr o of: By T heorem 6, if an (( n, K, R, d )) q subsystem code Q exists for K > 1 and 1 ≤ R < K , th en th ere exists an ad ditive code C ∈ F 2 n q and its su bcode D ≤ F 2 n q such t hat | C | = q n R/K and | D | = | C ∩ C ⊥ s | = q n /K R . Fur thermor e, d = min s wt( D ⊥ s \ C ) . Let w = ( w 1 , w 2 , . . . , w n ) and u = ( u 1 , u 2 , . . . , u n ) be tw o vectors in F n q . W .l.g., we can assume that the co de D ⊥ s is d efined as D ⊥ s = { ( u | w ) ∈ F 2 n q | w, u ∈ F n q } . Let w − 1 = ( w 1 , w 2 , . . . , w n − 1 ) and u − 1 = ( u 1 , u 2 , . . . , u n − 1 ) be two vector s in F n − 1 q . Also, let D ⊥ s p be th e co de o btained b y p unctur ing th e first coordin ate of D ⊥ s , hence D ⊥ s p = { ( u − 1 | w − 1 ) ∈ F 2 n − 2 q | w − 1 , u − 1 ∈ F n − 1 q } . since the minimum distance of D ⊥ s is at least 2, it follows that | D ⊥ s p | = | D ⊥ s | = K 2 | C | = K 2 q n R/K = q n RK and the minimu m distance of D ⊥ s p is at least d − 1 . Now , let us construct th e dual co de o f D ⊥ s p as f ollows. ( D ⊥ s p ) ⊥ s = { ( u − 1 | w − 1 ) ∈ F 2 n − 2 q | (0 u − 1 | 0 w − 1 ) ∈ D , w − 1 , u − 1 ∈ F n − 1 q } . Furthermo re, if ( u − 1 | w − 1 ) ∈ D p , then (0 u − 1 | 0 w − 1 ) ∈ D . Therefo re, D p is a self-orthog onal code a nd it has size g iv en by | D p | = q 2 n − 2 / | D ⊥ s p | = q n − 2 /RK . W e can also punc ture the co de C to the code C p at the first coordin ate, hence C p = { ( u − 1 | w − 1 ) ∈ F 2 n − 2 q | w − 1 , u − 1 ∈ F n − 1 q , ( aw − 1 | bu − 1 ) ∈ C, a, b ∈ F q } . Clearly , D ⊆ C and if a = b = 0 , th en the vector (0 u − 1 | 0 w − 1 ) ∈ D , the refore, ( u − 1 , w − 1 ) ∈ D p . This giv es us that D p ⊆ C p . Furth ermore, hen ce | C | = | C p | . The dual code C ⊥ s p can b e defined as C ⊥ s p = { ( u − 1 | w − 1 ) ∈ F 2 n − 2 q | w − 1 , u − 1 ∈ F n − 1 q , ( ew − 1 | f u − 1 ) ∈ C ⊥ s , e, f ∈ F q } . Also, if e = f = 0 , then D p ⊆ C ⊥ s p , furthermo re, D ⊥ s p = C p ∪ C ⊥ s p = { ( u − 1 | w − 1 ) ∈ F 2 n − 2 q | (2) (0 u − 1 | 0 w − 1 ) ∈ D } (3) Therefo re ther e exists a su bsystem code Q p = A p ⊗ B p . Also, the co de D ⊥ s p is pure an d has minimu m distan ce at least d − 1 . W e can proceed and compute the d imension of subsy stem A p and co -subsystem B p from Theo rem 6 as follows. (i) K p = q n − 1 / p | C p || D p | = q n − 1 / p ( q n R/K )( q n − 2 /RK ) = K , (ii) R p = ( | C p | / | D ′ p | ) 1 / 2 = (( q n R/K ) / ( q n − 2 /RK )) 1 / 2 = q R , (iii) d p = swt(( D p ) ⊥ s \ C p ) = swt(( D ⊥ s \ C p )) ≥ d − 1 . Therefo re, th ere exists a subsystem cod with parameter s (( n − 1 , K , q R, ≥ d − 1)) q . The m inimum distance conditio n follows since the cod e Q has d = min swt( D ⊥ s \ C ) and the code Q p has minimum distance as Q reduced by one. So, the minimum we ight of D ⊥ s p \ C p is at least the minimum weight o f ( D ⊥ s \ C ) − 1 d p = min swt( D p ⊥ s \ C p ) ≥ min swt( D ⊥ s \ C ) − 1 = d − 1 If the co de Q is pure, then min swt( D ⊥ s ) = d , there fore, the new code Q p is p ure since d p = min swt( D ⊥ s p ) ≥ d . W e con clude that if there is a subsystem code with param- eters (( n − 1 , K, q R, ≥ d − 1)) q , using [2, Theorem 2 .], th ere exists a code with p arameters (( n − 1 , q K, R, ≥ d − 1 )) q . 4 C. Reducing Dimension W e also ca n reduce dimension of the subsystem code f or fixed length n and minimu m d istance d , and still obtain a ne w subsystem code with improved minimum distance as sho wn in the fo llowing results. Theorem 12: If a (pure ) F q -linear [[ n, k , r, d ]] q subsystem code Q exists fo r d ≥ 2 , then there exists an F q -linear [[ n, k − 1 , r, d e ]] q subsystem code Q e (pure to d ) such tha t d e ≥ d . Pr o of: Existenc e of the [[ n, k , r, d ]] q subsystem cod e Q , implies existence of two add iti ve codes C ≤ F 2 n q and D ≤ F 2 n q such that | C | = q n − k + r and | D | = | C ∩ C ⊥ s | = q n − k − r . Furthermo re, d = min swt( D ⊥ s \ C ) and D ⊆ D ⊥ s . The idea of the p roof co mes by extendin g the code D by some vecto rs fro m D ⊥ s \ ( C ∪ C ⊥ s ). L et us choo se a code D e of size | q n +1 − r − k | = q | D | . W e also ensure that the code D e is self-orthogo nal. Clearly extending the code D to D e will extend both the codes C and C ⊥ s to C e and C ⊥ s e , respectively . Hence C e = q | C | = q n +1+ r − k and D e = C e ∩ C ⊥ s e . There exists a subsystem code Q e stabilized by the cod e C e . The result follows by com puting parameters o f the subsystem code Q e = A e ⊗ B e . (i) K e = q n / p | C e || D e | = q n / (( q n +1+ r − k )( q n +1 − k − r )) 1 / 2 = q k − 1 , (ii) R e = ( | C e | / | D e | ) 1 / 2 = (( q n +1 R/K ) / ( q n +1 /RK )) 1 / 2 = q r , (iii) d e = sw t(( D e ) ⊥ s \ C e ) ≥ swt(( D ⊥ s \ C e )) = d . If the inequality holds, th en th e code is pure to d . Arguably , It follows that the set ( D ⊥ s e \ C e ) is a subset of th e set D ⊥ s \ C b ecause C ≤ C e , he nce the m inimum weight d e is at least d . Lemma 1 3: Su ppose an [[ n, k , r, d ]] q linear pure su bsystem code Q exists generated by the two cod es C, D ≤ F 2 n q . Th en there exist linear [[ n − m, k ′ , r ′ , d ′ ]] q and [[ n − m, k ′ + r ′ − r ′′ , r ′′ , d ′ ]] q subsystem cod es with k ′ ≥ k − m , r ′ ≥ r , 0 ≤ r ′′ < k ′ + r ′ , and d ′ ≥ d for any integer m such that there exists a codeword of weight m in ( D ⊥ s \ C ) . Pr o of: [Sketch] This lemma 1 3 can be proved e asily by mapping the subsystem code Q into a stabilizer co de. By using [ 4, Theo rem 7.], a nd the ne w resulting stabilizer code can be map ped ag ain to a su bsystem code with the required parameters. D. Combining Subsystem Codes W e can also co nstruct new subsystem co des fro m given two sub system codes. The following theorem shows that two subsystem cod es can be merged tog ether into on e subsystem code with po ssibly impr oved distance or dimension . Theorem 14: Let Q 1 and Q 2 be two pure subsystem codes with par ameters [[ n 1 , k 1 , r 1 , d 1 ]] 2 and [[ n 2 , k 2 , r 2 , d 2 ]] 2 for k 2 + r 2 ≤ n 1 , respectively . The n ther e exists a subsy stem code with p arameters [[ n 1 + n 2 − k 2 − r 2 , k 1 + r 1 − r , r, d ]] 2 , where d ≥ min { d 1 , d 1 + d 2 − k 2 − r 2 } and 0 ≤ r < k 1 + r 1 . Pr o of: Existence of an [[ n i , k i , r i , d i ]] 2 pure su bsystem code Q i for i ∈ { 1 , 2 } , implies existence o f a p ure stabilizer code S i with parameters [[ n i , k i + r i , d i ]] 2 with k 2 + r 2 ≤ n 1 , see [2]. Therefore, by [4 , Th eorem 8.], the re exists a stabilizer code with par ameters [[ n 1 + n 2 − k 2 − r 2 , k 1 + r 1 , d ]] 2 , d ≥ min { d 1 , d 1 + d 2 − k 2 − r 2 } . Bu t th is cod e gives us a subsystem code with parameters [[ n 1 + n 2 − k 2 − r 2 , k 1 + r 1 − r , r , ≥ d ]] 2 with k 2 + r 2 ≤ n 1 and 0 ≤ r < k 1 + r 1 that proves the claim. Theorem 15: Let Q 1 and Q 2 be two pure subsy stem cod es with parameters [[ n, k 1 , r 1 , d 1 ]] q and [[ n, k 2 , r 2 , d 2 ]] q , respec- ti vely . If Q 2 ⊆ Q 1 , the n there exists an [[2 n, k 1 + k 2 + r 1 + r 2 − r , r , d ]] q pure sub system cod e with m inimum d istance d ≥ min { d 1 , 2 d 2 } an d 0 ≤ r < k 1 + k 2 + r 1 + r 2 . Pr o of: Existence of a pure subsystem code with parame- ters [[ n, k i , r i , d i ]] q implies existence o f a pure stab ilizer code with parameters [[ n, k i + r i , d i ]] q using [2, Theorem 4.]. But by using [5, Lemma 74.], there exists a pure stabilizer code with parameters [[2 n, k 1 + k 2 + r 1 + r 2 , d ]] q with d ≥ min { 2 d 2 , d 1 } . By [2, Th eorem 2., Coro llary 6 .], there mu st exist a pure sub- system co de with parameters [[2 n, k 1 + k 2 + r 1 + r 2 − r , r , d ]] q where d ≥ min { 2 d 2 , d 1 } and 0 ≤ r < k 1 + k 2 + r 1 + r 2 , which proves the claim. W e ca n recall the trace altern ativ e pro duct b etween two codewords of a classical code and the pr oof of T heorem 1 5 can b e stated as follows. Lemma 1 6: Let Q 1 and Q 2 be two p ure subsy stem codes with parameters [[ n, k 1 , r 1 , d 1 ]] q and [[ n, k 2 , r 2 , d 2 ]] q , respec- ti vely . If Q 2 ⊆ Q 1 , then there exists an [[2 n, k 1 + k 2 , r 1 + r 2 , d ]] q pure subsystem code with m inimum distance d ≥ min { d 1 , 2 d 2 } . Pr o of: Existence of the code Q i with parameter s [[ n, K i , R i , d i ]] q implies existence of two additive co des C i and D i for i ∈ { 1 , 2 } such that | C i | = q n R i /K i and | D i | = | C ∪ C ⊥ s | = q n /R i K i . W e k now that there exist ad ditive linear co des D i ⊆ D ⊥ a i , D i ⊆ C i , and D i ⊆ C ⊥ a i . Further more, D i = C i ∩ C ⊥ a i and d i = wt ( D ⊥ a i \ C i ) . Also, C i = q n + r i − k i and | D | = q n − r i − k i . Using the direct sum definition between to linear codes, let us construct a code D b ased o n D 1 and D 2 as D = { ( u, u + v ) | u ∈ D 1 , v ∈ D 2 } ≤ F 2 n q 2 . The code D h as size of | D | = q 2 n − ( r 1 + r 2 + k 1 + k 2 )= | D 1 || D 2 | . Also, we can define the cod e C based on the codes C 1 and C 2 as C = { ( a, a + b ) | a ∈ C 1 , b ∈ C 2 } ≤ F 2 n q 2 . The code C is of size | C | = | C 1 || C 2 | = q 2 n + r 1 + r 2 − k 1 − k 2 . But the trace- alternating dual of the code D is D ⊥ a = { ( u ′ + v ′ | , v ′ ) | u ′ ∈ D ⊥ a 1 , v ′ ∈ D ⊥ a 2 } . W e notice that ( u ′ + v ′ , v ′ ) is o rthogo nal to ( u, u + v ) because, from p roperties o f the pr oduct, h ( u, u + v ) | ( u ′ + v ′ , v ′ ) i a = h u | u ′ + v ′ i a + h u + v | v ′ i a = 0 holds for u ∈ D 1 , v ∈ D 2 , u ′ ∈ D ⊥ a 1 , and v ′ ∈ D ⊥ a 2 . 5 Therefo re, D ⊆ D ⊥ a is a self-o rthogo nal code with r espect to the trace alternating produ ct. Furthe rmore, C ⊥ a = { ( a ′ + b ′ , b ′ ) | a ′ ∈ C ⊥ a 1 , b ′ ∈ C ⊥ a 2 } . Hence, C ∩ C ⊥ a = { ( a, a + b ) ∩ ( aa + b ′ , b ′ ) } = D . T herefor e, there exists an F q -linear subsystem code Q = A ⊗ B with the following parameter s. i) K = | A | = q 2 n / ( | C || D | ) 1 / 2 = q 2 n p ( q 2 n R 1 R 2 /K 1 K 2 )( q 2 n /K 1 K 2 R 1 R 2 ) = q 2 n p q 2 n + r 1 + r 2 − k 1 − k 2 q 2 n − r 1 − r 2 − k 1 − k 2 = q k 1 k 2 = K 1 K 2 . ii) R = ( | C | | D | ) 1 / 2 = R 1 R 2 . iii) th e min imum distanc e is a direct consequen ce. Theorem 17: If there exist two pure subsystem qu antum codes Q 1 and Q 2 with parame ters [[ n 1 , k 1 , r 1 , d 1 ]] q and [[ n 2 , k 2 , r 2 , d 2 ]] q , respectiv ely . Then there exists a pur e sub sys- tem code Q ′ with parameters [[ n 1 + n 2 , k 1 + k 2 + r 1 + r 2 − r, r , ≥ min( d 1 , d 2 )]] q . Pr o of: T his Lemma can be proved easily fro m [2, The- orem 5.] and [5, Lemma 73.]. Th e idea is to map a pu re subsystem code to a pure stabilizer code, an d once again map the pure stabilizer cod e to a pure sub system cod e. Theorem 18: If there exist two pure subsystem qu antum codes Q 1 and Q 2 with parame ters [[ n 1 , k 1 , r 1 , d 1 ]] q and [[ n 2 , k 2 , r 2 , d 2 ]] q , respectively . Th en th ere exists a pure sub- system code Q ′ with parameters [[ n 1 + n 2 , k 1 + k 2 , r 1 + r 2 , ≥ min( d 1 , d 2 )]] q . Pr o of: Existence of the code Q i with parameter s [[ n, K i , R i , d i ]] q implies existence of two additive co des C i and D i for i ∈ { 1 , 2 } such th at | C i | = q n R i /K i and | D i | = | C ∪ C ⊥ s | = q n /R i K i . Let us c hoose the cod es C and D as follows. C = C 1 ⊕ C 2 = { uv | v ∈ C 1 , v ∈ C 2 } , and D = D 1 ⊕ D 2 = { ab | a ∈ D 1 , b ∈ C 2 } , respectively . From th is constru ction, and since D 1 and D 2 are self-o rthogo nal codes, it fo llows that D is also a self- orthog onal code. Fu rthermo re, D 1 ⊆ C 1 and D 2 ⊆ C 2 , then D 1 ⊕ D 2 ⊆ C 1 ⊕ C 2 , hence D ⊆ C . The code C is of size | C | = | C 1 || C 2 | = q ( n 1 + n 2 ) − ( k 1 + k 2 )+( r 1 + r 2 ) = q n 1 q n 2 R 1 R 2 /K 1 K 2 and D is of size | D | = | D 1 || D 2 | = q ( n 1 + n 2 ) − ( k 1 + k 2 ) − ( r 1 + r 2 ) = q n 1 q n 2 /R 1 R 2 K 1 K 2 . On th e other hand , C ⊥ s = ( C 1 ⊕ C 2 ) ⊥ s = C ⊥ s 2 ⊕ C ⊥ s 1 ⊇ D 2 ⊕ D 1 . Furthermo re, C ∩ C ⊥ s = ( C 1 ⊕ C 2 ) ∩ ( C ⊥ s 2 ∩ C ⊥ s 1 ) = D . Therefo re, there exists a subsystem code Q = A ⊗ B with the fo llowing parameters. i) K = | A | = q n 1 + n 2 / ( | C || D | ) 1 / 2 = q n 1 + n 2 p ( q n 1 + n 2 R 1 R 2 /K 1 K 2 )( q n 1 + n 2 /K 1 K 2 R 1 R 2 ) = q n 1 + n 2 p q n 1 + n 2 + r 1 + r 2 − k 1 − k 2 q n 1 + n 2 − r 1 − r 2 − k 1 − k 2 = q k 1 k 2 = K 1 K 2 = | A 1 || A 2 | . ii) R = ( | C | | D | ) 1 / 2 = s q n 1 q n 2 R 1 R 2 /K 1 K 2 q n 1 q n 2 /R 1 R 2 K 1 K 2 = R 1 R 2 = | B 1 || B 2 | . iii) th e min imum weight of D ⊥ s \ C is at least th e min imum weight of D ⊥ s 1 \ C 1 or D ⊥ s 2 \ C 2 . d = min { swt( D ⊥ s 1 \ C 1 ) , ( D ⊥ s 2 \ C 2 ) } ≥ min { d 1 , d 2 } . Theorem 19: Giv en two pure subsystem codes Q 1 and Q 2 with parameters [[ n 1 , k 1 , r 1 , d 1 ]] q and [[ n 2 , k 2 , r 2 , d 2 ]] q , respectively , with k 2 ≤ n 1 . An [[ n 1 + n 2 − k 2 , k 1 + r 1 + r 2 − r , r , d ]] q subsystem code exists su ch that d ≥ min { d 1 , d 1 + d 2 − k 2 } and 0 ≤ r < k 1 + r 1 + r 2 . Pr o of: T he p roof is a dir ect co nsequence as shown in the previous theorem s. Theorem 20: If an (( n, K, R , d )) q m pure subsystem co de exists, then there exists a p ure subsystem code with param eters (( nm, K, R , ≥ d )) q . Consequen tly , if a pure sub system co de with parameters (( nm, K , R , ≥ d )) q exists, then there exist a subsystem code with parame ters (( n, K , R, ≥ ⌊ d/m ⌋ )) q m .. Pr o of: Existenc e of a pu re subsystem cod e with p aram- eters (( n, K , R , d )) q m implies existence of a pur e stabilizer code with parameters (( n, K R, d )) q m using [2, Theor em 5 .]. By [5 , Lem ma 7 6.], th ere exists a s tabilizer code with param e- ters (( nm, K R, ≥ d )) q . From [2, The orem 2 ,5.], th ere exists a pure subsystem cod e with p arameters (( nm, K , R , ≥ d )) q that proves the first claim. By [5, L emma 76.] an d [2, T heorem 2,5.], and repeatin g th e same pro of, the second claim is a consequen ce. [t] I I I . S P E C I A L A N D S H O RT S U B S Y S T E M C O D E S [[8 , 1 , 2 , 3]] 2 A N D [[6 , 1 , 1 , 3 ]] 3 In this section we p resent the shortest subsystem co des over F 2 and F 3 fields. Theorem 5 implies that a stab ilizer code with parameters [[ n, k , d ]] q giv es subsystem co des with p arameters [[ n, k − r, r, d ]] q , see th e tables in [1]. 6 n \ k k-1 k k+1 n-1 [ r + 2 , d − 1] q [ ≤ r + 2 , d ] q , [ r + 1 , d − 1 ] q [ r , d − 1] q n [ r + 1 , d ] q , [ r + 1 , ≥ d ] q [ r , d ] q → [ ≤ r, ≥ d ] q [ r − 1 , d ] q → [ ≥ r, ≤ d ] q n+1 [ ≥ r, ≥ d ] q [ ≥ r, d ] q , [ r , ≥ d ] q Fig. 2. Existence of subsystem propagation rul es Consider a stabilize r cod e with par ameters [[8 , 3 , 3]] 2 . This code c an be used to derive [[8 , 2 , 1 , 3]] 2 and [[8 , 1 , 2 , 3]] 2 subsystem cod es. W e give an explicit co nstruction of th ese codes. Fur ther , we claim th at [[8 , 1 , 2 , 3]] 2 and [[8 , 2 , 1 , 3]] 2 are the shortest no ntrivial binary subsystem codes. W e show the stabilizer and no rmalizer matrice s fo r these codes. Also, we prove their minimum distan ces using the weight enumeratio n of these codes. W e present tw o codes with less len gth, howe ver we can not tolerate mor e than 2 ga uge qubits. Th e following example shows [[8 , 1 , 2 , 3]] sho rt sub system co de over F 2 . Example 21: D S =       X I Y I Z Y X Z Y I Y X I Z Z X I X Y Y Z X Z I I Y I Z Y X X Z I I X Z X Y Z Y       (4) D ⊥ S =                   X I I I I I Z Y Y I I I I Y X X I X I I I Y Y X I Y I I I I X Z I I X I I Y Z I I I Y I I I Z X I I I X I Y I Z I I I Y I Y Y Y I I I I X I Y Z I I I I Y Y Z Z I I I I I Z X Y                   (5) C S =           X I Y I Z Y X Z Y I Y X I Z Z X I X Y Y Z X Z I I Y I Z Y X X Z I I X Z X Y Z Y Y I I I I Y X X I X I I I Y Y X           (6) C ⊥ S =           X I Y I Z Y X Z Y I Y X I Z Z X I X Y Y Z X Z I I Y I Z Y X X Z I I X Z X Y Z Y X I I I I I Z Y I I I Y I Y Y Y           (7) W e notice that th e matrix D S generates the cod e D = C ∩ C ⊥ s . Further more, dimensions o f the sub systems A and B ar e giv en by k = dim D ⊥ s − dim C = (11 − 7) / 2 = 2 an d r = dim C − dim D = (7 − 5) / 2 = 1 . Hence we ha ve [[8 , 2 , 1 , 3]] 2 and [[8 , 1 , 2 , 3 ]] 2 subsystem codes. W e show that th e subsystem co des [[8 , 1 , 2 , 3]] 2 is not better than the s tabilizer code [[8 , 3 , 3]] 2 in ter ms of syndrome measuremen t. T he reason is that the form er needs 8 − 1 − 2 = 5 syndrom e me asurements, while the later n eeds also 8 − 3 = 5 measuremen ts. This is an obviou s example wher e sub system codes ha ve no superiority in terms of syndrome measurements. W e post an o pen qu estion r egarding th e thresho ld value and fault tolerant gate oper ations f or this co de. W e do not k now at this time if the cod e [[8 , 1 , 2 , 3]] 2 has better threshold value and less fault-to lerant op erations. Also , do es the subsystem code with par ameters [[8 , 1 , 3 , 3 ]] 2 exist? No nontrivial [[7 , 1 , 1 , 3]] 2 exists. Th ere exists a tri vial [[7 , 1 , 1 , 3]] 2 code obtained by simply e xtending the [[7 , 1 , 3]] 2 code as the [[5 , 1 , 3]] 2 code. W e show the smallest subsystem code with leng th 7 must have at most m inimum weight equals to 2. Since [[7 , 2 , 2]] 2 exists, then we can construct the stabilizer an d n ormalizer matrices as follows. D S =       X X X X I I I Y Y Y Y I I I I I I I X I I I I I I I X I I I I I I I X       (8) D ⊥ S =               X I I X I I I Y I I Y I I I I X I X I I I I Y I Y I I I I I X X I I I I I Y Y I I I I I I I X I I I I I I I X I I I I I I I X               (9) Clearly , from o ur construction and using Theorem 5, there must exist a sub system code with p arameters k an d r given as follows. dim D ⊥ s = 9 / 2 and dim C = 7 / 2 . A lso, dim D = 5 / 2 and min ( D ⊥ s \ C ) = 2 . Theref ore, , k = (9 − 7) / 2 = 1 and r = (7 − 5) / 2 = 1 . Conseque ntly , the p arameters o f the subsystem code ar e [[7 , 1 , 1 , 2]] 2 . This example shows [[6 , 1 , 1 , 3]] short subsy stem code over F 3 . Example 2 2: W e g iv e a non trivial short sub system code over F 3 . This is derived fr om th e [[6 , 2 , 3]] 3 graph q uantum 7 code. Also, we show in [1] an example for an [[6 , 1 , 1 , 3]] 7 subsystem code over F 7 . Con sider th e field F 3 and let C ⊆ F 12 3 be a linear code defin ed by the following generator matrix. C =         1 0 0 0 2 0 0 2 0 2 0 2 0 1 0 0 0 2 1 0 1 0 1 0 0 0 1 0 2 0 0 1 0 1 0 1 0 0 0 1 0 2 2 0 2 0 2 0 0 0 0 0 1 0 0 2 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0         =   S X 1 Z 1   . Let the symplectic inner pr oduct h ( a | b ) | ( c | d ) i s = a · d − b · c . Then the symp lectic d ual o f C is gener ated by C ⊥ s =   S X 2 Z 2   , where X 2 =  0 0 0 0 0 1 1 0 2 0 0 0  and Z 2 =  0 0 0 0 0 0 0 1 0 1 0 1  . T he matrix S g enerates the code D = C ∩ C ⊥ s . Now D defines a [[6 , 2 , 3 ]] 3 stabilizer code. Theref ore, s wt( D ⊥ s \ D ) = 3 . It fol- lows that swt( D ⊥ s \ C ) ≥ s wt( D ⊥ s ) = 3 . By [3, Theor em 4], we have a [[6 , (dim D ⊥ s − dim C ) / 2 , (dim C − dim D ) / 2 , 3]] 3 viz. a [[6 , 1 , 1 , 3]] 3 subsystem code. W e can also have a tri vial [[6 , 1 , 1 , 3]] 2 code. T his trivial extension seems to argue against th e usefu lness of subsystem codes an d if th ey will really lead to improvement in perfor- mance. An obvious o pen question is if there exist n ontrivial [[6 , 1 , 1 , 3]] 2 or [[7 , 1 , 1 , 3]] 2 subsystem codes. R E F E R E N C E S [1] S. A. Aly and A. Klappenec ker . Struc tures and constructions of subsysem codes ov er finite fiel ds. Phys. Re v . A. , 2008. on submissio n. [2] S. A. Aly an d A. Klappene cke r . Subsyste m code construc tions. In Pr oc. 2008 IEEE International Symposium on Information Theory , T or onto, Canada , Submitted, 2008. [3] S. A. Aly , A. Klappeneck er , and P . K. Sarvepalli . Subsystem codes. 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