The existence of a quantum $[[28,12,6]]$ code was one of the few cases for codes of length $n\le 30$ that was left open in the seminal paper by Calderbank, Rains, Shor, and Sloane \cite{CRSS}. The main result of this paper is the construction of a new optimal linear quaternary $[28,20,6]$ code which contains its hermitian dual code and yields an optimal linear quantum $[[28,12,6]]$ code

Tonchev, Vladimir D.

English

Journal of Algebra Combinatorics Discrete Structures and Applications (JACODESMATH, Yildiz Technical University - YTU)

JACODESMATH / ISSN 2148-838XJ. Algebra Comb. Discrete Appl.1(1) • 13-17Received: 31 May 2014; Accepted: 3 August 2014DOI 10.13069/jacodesmath.25090Journal of Algebra Combinatorics Discrete Structures and ApplicationsThe existence of optimal quaternary [28, 20, 6] andquantum [[28, 12, 6]] codesResearch ArticleVladimir D. Tonchev ∗Michigan Technological University, Houghton, Michigan 49931-1295, USAAbstract: The existence of a quantum [[28, 12, 6]] code was one of the few cases for codes of length n ≤ 30 thatwas left open in the seminal paper by Calderbank, Rains, Shor, and Sloane [2]. The main result ofthis paper is the construction of the first optimal linear quaternary [28, 20, 6] code which contains itsHermitian dual code and yields the first optimal quantum [[28, 12, 6]] code.2010 MSC: 94B05, 94B65, 81-99Keywords: Quantum code, Optimal code1. IntroductionWe assume familiarity with the basics of classical and quantum error-correcting codes [2], [5].The Hermitian inner product in GF (4)n is defined as(x, y)H =n∑i=1xiy2i , (1)while the trace inner product in GF (4)n is defined as(x, y)T =n∑i=1(xiy2i + x2i yi). (2)A code C is self-orthogonal if C ⊆ C⊥, and self-dual if C = C⊥. A linear code C ⊆ GF (4)n is self-orthogonal with respect to the trace product (2) if and only if it is self-orthogonal with respect to theHermitian product (1) [2].∗ E-mail: tonchev@mtu.edu13The existence of optimal quaternary [28, 20, 6] and quantum [[28, 12, 6]] codesAn additive (n, 2k) code C over GF (4) is a subset of GF (4)n consisting of 2k vectors which is closedunder addition. An additive code is even if the weight of every codeword is even, and otherwise odd.Note that an even additive code is trace self-orthogonal, and a linear self-orthogonal code is even [2]. IfC is an (n, 2k) additive code with weight enumeratorW (x, y) =n∑j=0Ajxn−jyj , (3)the weight enumerator of the trace-dual code C⊥ is given byW⊥ = 2−kW (x+ 3y, x− y) (4)In their seminal paper [2], Calderbank, Rains, Shor and Sloane described a method for the construc-tion of quantum error-correcting codes from additive codes that are self-orthogonal with respect to thetrace product (2).Theorem 1.1. [2] An additive trace self-orthogonal (n, 2n−k) code C such that there are no vectors ofweight < d in C⊥ \ C yields a quantum code with parameters [[n, k, d]].A quantum code associated with an additive code C is pure if the minimum distance of C⊥ is d;otherwise, the code is called impure. A quantum code is called linear if the associated additive code C islinear.A table with lower and upper bounds on the minimum distance d for quantum [[n, k, d]] codes oflength n ≤ 30 is given in the paper by Calderbank, Rains, Shor and Sloane [2]. In particular, accordingto Table III on page 1382 in [2], the largest minimum distance d of a known quantum [[28, 12]] code isd = 5, while the best upper bound is d ≤ 6. In the next section, we describe a simple construction ofquaternary Hermitian self-orthogonal codes with parameters [2n + 1, k + 1] and [2n + 2, k + 2] from agiven pair of Hermitian self-orthogonal [n, k] codes. As an application of this construction, we find thefirst optimal quaternary linear [28, 20, 6] which contains its dual code and hence yields the first optimal[[28, 12, 6]] quantum code.An extended version of Calderbank-Rains-Shor-Sloane table for quantum codes [2, Table III], as wellas tables with bounds on the minimum distance of linear codes, was compiled by Grassl [4].2. A doubling constructionLemma 2.1. Suppose that Ci (i = 1, 2) is a linear Hermitian self-orthogonal [n, k] code over GF (4)with generator matrix Gi, and x(i) ∈ C⊥i is a vector of odd weight.(a) The code C ′ with generator matrixG′ =0G1 G2 . . .0x(1) 0 . . . 0 1 (5)is a Hermitian self-orthogonal [2n+ 1, k + 1] code with dual distanced(C ′)⊥ ≤ min(d(C⊥11), d(C⊥2 )), (6)where C11 is the code spanned by the rows of G11 given by (7):G11 =0G1 . . .0x(1) 1 . (7)14V. D. Tonchev(b) The code C ′′ with generator matrixG′′ =0 0G1 G2 . . .0 0x(1) 0 . . . 0 1 00 . . . 0 x(2) 0 1 (8)is a Hermitian self-orthogonal [2n+ 2, k + 2] code with dual distanced(C ′′)⊥ ≤ min(d(C⊥11), d(C⊥22)), (9)where C22 is the code spanned by the rows of G22 given by (10):G22 =0G2 . . .0x(2) 1 . (10)Proof. The self-orthogonality of C ′ and C ′′ follows from the fact that all rows of G′ and G′′ haveeven weights, and every pair of rows of G′, as well as every pair of rows of G′′, are pairwise orthogonal.Since the weight of x(1) (resp. x(2)) is odd, x(1) does not belong to C1, and x(2) does not belong to C2,and that implies the dimensions of C ′ and C ′′. The bounds (6), (9) on the dual distance follow triviallyby the observation that every codeword of C⊥11 (resp. C⊥22) extends to a codeword of (C ′)⊥ (resp (C ′′)⊥)by filling in all remaining coordinates with zeros. It is worth mentioning that since C1 and C2 are self-orthogonal, their minimum distances are trivialupper bounds on the minimum dual distances d(C ′)⊥ and d(C ′′)⊥. For example, if d(C1) = 2 thend(C ′)⊥ ≤ 2.We note also that using codes C1, C2 with large minimum distances is a necessary, but not alwayssufficient condition for large dual distances d(C ′)⊥ and d(C ′′)⊥. For example, if G1 = G2 and x(1) is theall-one vector, then for every 1 ≤ i ≤ n, the columns of (5) with indices i, i+ n and 2n+ 1 determine acodeword of weight 3 in (C ′)⊥.These simple observations illustrate that one cannot expect a good general lower bound on d(C ′)⊥or d(C ′′)⊥, and finding codes C1, C2 with appropriate generator matrices G1, G2 and vectors x(1), x(2)which lead to optimal dual distances d(C ′)⊥ and d(C ′′)⊥ is not a trivial task.Using the connection to quantum codes described in Theorem 1.1, Lemma 2.1 implies the following.Corollary 2.2. The existence of quaternary Hermitian self-orthogonal [n, k] codes Ci (i = 1, 2) satisfyingthe assumptions of Lemma 2.1 implies the existence of a pure quantum linear [[2n + 1, 2n − 2k − 1, d′]]code with d′ ≤ min(d(C⊥11), d(C⊥2 )), and a pure quantum linear [[2n + 2, 2n − 2k − 2, d′′]] code withd′′ ≤ min(d(C⊥11), d(C⊥22)).We will apply Lemma 2.1 and Corollary 2.2 to some self-orthogonal codes of length n = 2k + 1being shortened codes of extremal self-dual [2k+2, k+1] codes, that is, self-dual codes having maximumpossible minimum distance for the given code length.Example 2.3. The matrixG1 =(1 0 1 ω ω0 1 ω ω 1)is the generator matrix of a self-orthogonal [5, 2, 4] code C1 over GF (4) = {0, 1, ω, ω2}. The code C1is a shortened code of the unique (up to equivalence) self-dual [6, 3, 4] code. Applying Lemma 2.1 with15The existence of optimal quaternary [28, 20, 6] and quantum [[28, 12, 6]] codesC2 = C1, G2 = G1, and x(1) = x(2) being the all-one vector of length 5, gives a self-orthogonal [11, 3] codeC ′ with dual distance 3 and a self-orthogonal [12, 4] code C ′′ with dual distance 4, which gives optimalquantum [[11, 5, 3]] and [[12, 4, 4]] codes respectively via Corollary 2.2.Example 2.4. A pair of self-orthogonal [7, 3] codes obtained as shortened codes of the unique (up toequivalence) self-dual [8, 4, 4] code can be used to obtain optimal quantum [[15, 7, 3]] and [[16, 6, 4]] codes.3. An optimal quantum [[28, 12, 6]] codeThe smallest parameters of a self-dual quaternary linear code that yields a quantum code withminimum distance d ≥ 5 via Corollary 2.2 are [14, 7, 6]. The only such code, up to equivalence, is thequaternary extended quadratic residue code q14 [6, page 340]. We apply Lemma 2.1 using the pair ofself-orthogonal [13, 6] codes C1, C2 generated by the following matrices:G1 =000010021023330000100210233300001002102233000010021002330000100211023300001002, G2 =000011302300220000113023000200001130230002000011302330020000113022300200001130,where for convenience, the elements ω and ω2 of GF (4) are written as 2 and 3 respectively. The matricesG1, G2 are circulant. The codes C1, C2 are cyclic and equivalent to a shortened code of q14.Choosing x(1) = x(2) to be the all-one vector of length 13, we obtain the generator matrix G′ (5) of aself-orthogonal [27, 7] code C ′ with dual distance 5, and the generator matrix G′′ (8) of a self-orthogonal[28, 8] code with dual distance 6. The matrix G′′ is available on line athttp://www.math.mtu.edu/~tonchev/gm28-8.htmlBy Corollary 2.2, C ′ gives a pure optimal quantum [[27, 13, 5]] code, while C ′′ gives a pure optimalquantum [[28, 12, 6]] code.An alternative geometric construction of a quantum code with the first parameters, [[27, 13, 5]], wasgiven by the author in [7]. To the best of our knowledge, the quantum code with the second parameters,[[28, 12, 6]], is the first known quantum code with these parameters (a quantum [[28, 12, 5]] code was listedin [2]).The weight distribution of the [28, 8] code C ′′ is given in Table 1.The weight enumerator of the dual [28, 20] code (C ′′)⊥ is1 + 6240y6 + 37128y7 + 314223y8 + 2044848y9 + 11883768y10 + . . .We note that the code (C ′′)⊥ is an optimal linear [28, 20, 6] quaternary code: 6 is the best theoreticalupper bound on the minimum distance of a quaternary linear [28, 20] code. The largest minimum distanceof any previously known [28, 20] code was 5 [3], [4].AcknowledgmentsMagma [1] was used for some of the computations. This research was partially supported by NSAGrant H98230-12-0213. The author wishes to thank the unknown reviewers for noticing some typos andsuggesting several improvements of the text layout and notation.16V. D. TonchevTable 1.w Aw12 3914 616 319818 920420 1821322 2285424 1056926 124828 204References[1] W. Bosma, J. Cannon, J, Handbook of Magma Functions, Department of Mathematics, University ofSydney, 1994.[2] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codesover GF (4), IEEE Trans. Information Theory, 44(4), 1369-1387, 1998.[3] A. E. Brouwer, Tables of linear codes, http://www.win.tue.nl/ aeb/.[4] M. Grassl, http://www.codetables.de.[5] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland,Amsterdam 1977.[6] G. Nebe, E. M. Rains, N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.[7] V. D. Tonchev, Quantum codes from caps, Discrete Math., 308, 6368-6372, 2008.17

The existence of optimal quaternary [28,20,6] and quantum [[28,12,6]] codes

https://jacodesmath.com/index.php/jacodesmath/article/download/3/2

The existence of optimal quaternary [28,20,6] and quantum [[28,12,6]] codes

Abstract

Similar works

Full text

Available Versions

Journal of Algebra Combinatorics Discrete Structures and Applications (JACODESMATH, Yildiz Technical University - YTU)