In this paper, we establish the necessary and sufficient conditions for quasi-cyclic (QC) codes with index even to be symplectic self-orthogonal. Subsequently, we present the lower and upper bounds on the minimum symplectic distances of a class of 1-generator QC codes and their symplectic dual codes by decomposing code spaces. As an application, we construct many new binary symplectic self-orthogonal QC codes with excellent parameters, leading to 117 record-breaking quantum error-correction codes.

The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codeshttps://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.

Let$q$be a prime power and$m\geq 3$be odd. Suppose that$n=\frac {q^{2m}-1}{a}$with$a|(q^{m}+1)$and$3\leq a \leq 2(q^{2}-q+1)$. This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over$\mathbb {F}_{q^{2}}$of length$n$. Firstly, we give the maximum designed distance$\delta _{m,a}^{R}$of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to$\delta _{m,a}^{R}$. It is worth mentioning that our maximum designed distance$\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}$, where$\delta _{m}^{A}$is given by Alyet al.(IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature.

In this correspondence, we use a symplectic geometry over the binary field to discuss the equivalence of additive codes over the quaternary field and the equivalence of additive quantum codes. We establish the existence of a standard form of the stabilizer and the normalizer matrices for additive quantum codes. Thus, we present the quantum analogue of standard forms of generator and parity check matrices for systematic linear codes in classical coding theory for additive quantum codes.

The classification of binary [n,k,d] codes withdgess2k-1 and without zero coordinates is reduced to the classification of binary [(2k-1)c(k,s,t)+t,k,d] code forn=(2k-1)s+t,sges 1 and 1 lestles 2k-2, wherec(k,s,t) les min{s,t} is a function ofk,s, andt. 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 fortisin {1,2,6,7,8,9,13,14} andsges 1 is obtained, and the generator matrices and weight polynomials of these 4-dimensional optimal self-orthogonal codes are also given.

We give elementary recursive constructions of binary self-orthogonal codes with dual distance four for all even lengths n/spl ges/12 and n=8. Consequently, good quantum codes of minimum distance three and four for such length n are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before.