Related papers: The ${[46,9,20]_2}$ code is unique

All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…

It is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2; thus, it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize. With…

For $n,d,w \in \mathbb{N}$, let $A(n,d,w)$ denote the maximum size of a binary code of word length $n$, minimum distance $d$ and constant weight $w$. Schrijver recently showed using semidefinite programming that $A(23,8,11)=1288$, and the…

Using the method for constructing binary self-dual codes with an automorphism of order square of a prime number we have classified all binary self-dual codes with length 76 having minimum distance $d=14$ and automorphism of order 9. Up to…

Let $t \in \{2,8,10,12,14,16,18\}$ and $n=31s+t\geq 14$, $d_{a}(n,5)$ and $d_{l}(n,5)$ be distances of binary $[n,5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n,5,d_{a}(n,5)]$…

Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In…

Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$,…

We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size…

The existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a…

It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$…

Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was…

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main…

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing Solomon-Stiffler codes and some related residual codes. Second, using such a…

We classify binary completely regular codes of length $m$ with minimum distance $\delta$ for $(m,\delta)=(12,6)$ and $(11,5)$. We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard…

In this paper, we construct a new family of distance-optimal binary cyclic codes with the minimum distance $6$ and a new family of distance-optimal quaternary cyclic codes with the minimum distance $4$. We also construct several families of…

In this article, we construct linear codes over the commutative non-unital ring $I$ of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are…

An additive quaternary $[n,k,d]$-code (length $n,$ quaternary dimension $k,$ minimum distance $d$) is a $2k$-dimensional F_2-vector space of $n$-tuples with entries in $Z_2\times Z_2$ (the $2$-dimensional vector space over F_2) with minimum…

The hull of a linear code over finite fields is the intersection of the code and its dual, and linear codes with small hulls have applications in computational complexity and information protection. Linear codes with the smallest hull are…

Kim et al. (2021) gave a method to embed a given binary $[n,k]$ code $\mathcal{C}$ $(k = 3, 4)$ into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. We extend this…