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Let $ d_{a}(n, 5) $ and $ d_{l}(n, 5) $ be the minimum weights of optimal binary $ [n, 5] $ linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate $ d_{l}(n, 5) $ of some families of binary $ [n, 5] $ LCD codes when $ n = 31s+t\geq 14 $ with $ s $ integer and $ t \in\; \{2, 8, 10, 12, 14, 16, 18\} $. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary $ [n, 5, d_{a}(n, 5)] $ LCD codes is verified, and we further derive that $ d_{l}(n, 5) = d_{a}(n, 5)-1 $ for $ t\neq 16 $ and $ d_{l}(n, 5) = 16s+6 = d_{a}(n, 5)-2 $ for $ t = 16 $. Combining them with known results on optimal LCD codes, $ d_{l}(n, 5) $ of all $ [n, 5] $ LCD codes are completely determined.

Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.

We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.

Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n≤25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl’s code table, and some of them are nearly optimal.

In this paper, we consider S-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new S-chain search method. As computational results, 18 S-chains with large distances are obtained, and many good quantum codes can be derived from those S-chains by Steane construction, some of which improve the previous results.

Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. A linear code $\mathcal {C}$ is said to have $(r,\delta)$ -locality if for each coordinate $i$ , there exists a punctured subcode of $\mathcal {C}$ with support containing $i$ , whose length is at most $r+\delta -1$ , and whose minimum distance is at least $\delta $ . An LRC is called optimal if its minimum distance attains Singleton-type bound was proposed. In this letter, optimal LRCs are considered. We first determine $(r,\delta)$ -locality of three dimensional Simplex code, then using anticode strategy, a class of $[{3q,3,2q-1}]_{q}$ LRCs with $(2,q)$ locality are derived for general $q$ . Finally, using an ovoid in $PG(3, q)$ , we construct $[q^{2}+1,4,q(q-1)]_{q}$ and $[{4q-4,4,3q-5}]_{q}$ LRCs with $r=3$ and $\delta =q-1$ . All LRCs constructed in this letter attain the Singleton-type bound.

In this short paper, our main objective is to construct binary locally repairable codes (LRCs) with good properties. Two constructions of LRCs with short lengthes are proposed. The first one is that define a $u$ -linearly independent set ( $u$ -LIS) of an LRC with disjoint repair groups (DRGs) and enlarge it into another one with bigger size to construct new LRCs. The second is puncturing check matrices of known codes to construct new LRCs. As an application, many new binary LRCs are constructed from distance optimal linear codes, which are also locality optimal according to the Cadambe-Mazumdar (C-M) bound.

In this paper, we present a computer-supported method of searching for quantum caps. By means of this method and relevant knowledge of combinatorics, many quantum caps in $PG(3,9)$ and $PG(4,9)$ are constructively proven to exist. Then, according to the theorem that each quantum cap corresponds to a quantum error-correcting code with $d=4$ , we obtain 278 quantum error-correcting codes. Most of these results break the GV bound, and a number of them are optimal quantum codes or have improved parameters.

In this paper, we consider a family of one-generator quasi-cyclic (QC) codes and their applications in quantum codes construction. We give a sufficient condition for one-generator QC codes to be self-orthogonal with respect to the symplectic inner product. As the computational results, five new binary quantum codes with parameters [[45,29,5]], [[63,36,7]], [[73,55,5]], [[105,71,7]], and [[105,72,7]] improving the best-known lower bounds on minimum distance in Grassl's code tables are constructed.

In this paper, from special code chains $\mathcal {C}_{1} \supseteq \mathcal {C}_{2}~\supseteq \mathcal {C}_{3}$ such that $\mathcal {C}^{\bot _{h}}_{1}\subseteq \mathcal {C}_{3}$ and $\mathcal {C}^{\bot _{h}}_{2}\subseteq \mathcal {C}_{2}$ , some Hermitian dual-containing (HDC) matrix-product (MP) codes are presented, where $\mathcal {C}_{3}$ is not HDC. By studying some constacyclic codes of lengths $n=q^{2}\pm 1$ and $n=\frac {q^{2}-1}{2}$ , we construct many HDC MP codes of length $3n$ . Consequently, many $q$ -ary quantum codes with larger designed distance $d\geq q+1$ are obtained from these MP codes, where $4\leq q\leq 9$ .