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We present two constructions for binary self-orthogonal codes. It turns out that our constructions yield a constructive bound on binary self-orthogonal codes. In particular, when the information rate R=1/2, by our constructive lower bound,…
For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight…
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 present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of…
In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring…
Linear codes are widely studied due to their applications in communication, cryptography, quantum codes, distributed storage and many other fields. In this paper, we use the trace and norm functions over finite fields to construct a family…
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…
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…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, a class of $q$-ary linear codes with few weights are presented and their weight distributions are determined using Gauss periods. Some of…
This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the…
Let $F_2$ be the binary field and $Z_{2^r}$ the residue class ring of integers modulo $2^r$, where $r$ is a positive integer. For the finite $16$-element commutative local Frobenius non-chain ring $Z_4+uZ_4$, where $u$ is nilpotent of index…
Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight Griesmer $[\frac{q^k-1}{q-1},k…
We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and…
In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However,…
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…
We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R=\mathbb{F}_2 +v\mathbb{F}_2 +v^2\mathbb{F}_2 +v^3\mathbb{F}_2 +v^4\mathbb{F}_2,$ where $v^5=1.$ The same ring occurs in the quintic…
In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring…
Most bounds on the size of codes hold for any code, whether linear or not. Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper we identify code parameters…
This text contains some notes on the Griesmer bound. In particular, we give a geometric proof of the Griesmer bound for the generalized weights and show that a Solomon--Stiffler type construction attains it if the minimum distance is…