Optimal Self-orthogonal Codes Research Articles

On construction of ternary optimal self-orthogonal codes

On construction of ternary optimal self-orthogonal codes

Embedding Linear Codes Into Self-Orthogonal Codes and Their Optimal Minimum Distances

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ( k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ). For k > 4, we suggest a recursive method to embed a k-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n \not ≡ 6, 13,14,21,22,28,29 (mod 31) with dimension 5.

On Euclidean self-dual codes and isometry codes

In this paper, we provide new methods and algorithms to construct Euclidean self-dual codes over large finite fields. With the existence of a dual basis, we study dual preserving linear maps, and as an application, we use them to construct self-orthogonal codes over small finite prime fields using the method of concatenation. Many new optimal self-orthogonal and self-dual codes are obtained.

New binary quantum stabilizer codes from the binary extremal self-dual $$[48, 24, 12]$$ [ 48 , 24 , 12 ] code

This paper is devoted to constructing binary quantum stabilizer codes based on the binary extremal self-dual code of parameters $$[48, 24, 12]$$[48,24,12] by Steane's construction. First, we provide an explicit generator matrix for the unique self-dual $$[48,24,12]$$[48,24,12] code to see it as a one-generator quasi-cyclic one and obtain six optimal self-orthogonal codes of parameters $$[48 - t, 24 - t, 12]$$[48-t,24-t,12] for $$1 \le t \le 6$$1≤t≤6 with dual distances from 11 to 7 by puncturing the $$[48,24,12]$$[48,24,12] code. Second, a special type of subcode structures for self-orthogonal codes is investigated, and then ten derived dual chains are designed. Third, twelve binary quantum codes are constructed from these derived dual pairs within dual chains using Steane's construction. Ten of them, $$[[42,10,8]], [[44,11,8]], [[45,10,8]], [[45,8,9]], [[46,12,8]]$$[[42,10,8]],[[44,11,8]],[[45,10,8]],[[45,8,9]],[[46,12,8]], $$[[46,9,9]], [[46,5,10]], [[48,13,8]], [[48,9,9]]$$[[46,9,9]],[[46,5,10]],[[48,13,8]],[[48,9,9]], and $$[[48,4,11]]$$[[48,4,11]], achieve as good parameters as the best known ones with comparable lengths and dimensions. Two other codes of parameters $$[[47,4,11]]$$[[47,4,11]] and $$[[48,3,12]]$$[[48,3,12]] are record breaking in the sense that they improve on the best known ones with the same lengths and dimensions in terms of distance.

Optimal binary codes and binary construction of quantum codes

This paper discusses optimal binary codes and pure binary quantum codes created using Steane construction. First, a local search algorithm for a special subclass of quasi-cyclic codes is proposed, then five binary quasi-cyclic codes are built. Second, three classical construction methods are generalized for new codes from old such that they are suitable for constructing binary self-orthogonal codes, and 62 binary codes and six subcode chains of obtained self-orthogonal codes are designed. Third, six pure binary quantum codes are constructed from the code pairs obtained through Steane construction. There are 66 good binary codes that include 12 optimal linear codes, 45 known optimal linear codes, and nine known optimal self-orthogonal codes. The six pure binary quantum codes all achieve the performance of their additive counterparts constructed by quaternary construction and thus are known optimal codes.

Classification of Quaternary [21s+17,3] Optimal Self-orthogonal Codes

Classification of Quaternary [21s+17,3] Optimal Self-orthogonal Codes

On The Classification of Binary Optimal Self-Orthogonal Codes

The classification of binary [n,k,d] codes with d ges s2k-1 and without zero coordinates is reduced to the classification of binary [(2k-1)c (k,s,t)+t,k,d] code for n =(2k-1)s+t, s ges 1 and 1 les t les 2k-2, where c(k,s,t) les min{s, t} is a function of k, s, and t. Binary [15s +t, 4] optimal self-orthogonal codes are characterized by systems of linear equations. Based on these two results, the complete classification of [15s +t,4] optimal self-orthogonal codes for t isin {1,2,6,7,8,9,13,14} and s ges 1 is obtained, and the generator matrices and weight polynomials of these 4-dimensional optimal self-orthogonal codes are also given.

Some optimal self-orthogonal and self-dual codes

The binary optimal self-orthogonal codes of dimension 4 ⩽ k ⩽ 10 and length 15 ⩽ n ⩽ k + 16 are classified. They are used for constructing self-dual codes of lengths 40, 42, and 44. All optimal binary self-dual codes of lengths 42 and 44 which have an automorphism of order 5 with four independent cycles are obtained up to equivalence. All optimal self-dual codes of lengths 40, 42 and 44 which have an automorphism of order 3 with six independent cycles are constructed. Some of them are the first known codes with their weight enumerators.

New results on self-orthogonal unequal error protection codes

A lower bound on the length of binary self-orthogonal unequal error protection (UEP) codes is derived, and two design procedures for constructing optimal self-orthogonal UEP codes are proposed. With this lower bound, known self-orthogonal UEP codes can be evaluated. It is pointed out that, for given values of minimum distance and code rate, the self-orthogonal codes must be relatively long, so optimal self-orthogonal codes are not optimal in general. But self-orthogonal codes can be implemented simply, and they have error-correcting capabilities beyond those guaranteed by their minimum distance. These properties can be viewed as a partial compensation for using self-orthogonal codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>