Related papers: Some Bounds on Binary LCD Codes
We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. In turn, these lead to two new fundamental…
Linear codes for error detection on a q-ary symmetric channel are studied. It is shown that for given dimension k and minimum distance d, there exists a value \mu(d,k) such that if C is a code of length n >= \mu(d,k), then neither C nor its…
For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no…
We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent then they are additively equivalent.
A subspace code is a nonempty collection of subspaces of the vector space $\mathbb{F}_q^{n}$. A pair of linear codes is called a linear complementary pair (in short LCP) of codes if their intersection is trivial and the sum of their…
The hull of a linear code $C$ is the intersection of $C$ with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension $\ge 2$ except for self-orthogonal codes. We propose a…
A linear code is said to be self-orthogonal if it is contained in its dual. Self-orthogonal codes are of interest because of their important applications, such as for constructing linear complementary dual (LCD) codes and quantum codes. In…
This work explores LCD and self-dual codes over a noncommutative non-unital ring $ E_p= \langle r,s ~|~ pr =ps=0,~ r^2=r,~ s^2=s,~ rs=r,~ sr=s \rangle$ of order $p^2$ where $p$ is a prime. Initially, we study the monomial equivalence of two…
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 $\mathbb{F}_q$…
In this paper, we introduce a general construction of linear codes with small dimension hull from any non LCD codes. Furthermore, we show that for any linear code $\Co$ over $\F_q$ ($q > 3$) with $dim(Hull(\Co))=h$ there exist an equivalent…
The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this…
The existence of $q$-ary linear complementary pairs (LCPs) of codes with $q> 2$ has been completely characterized so far. This paper gives a characterization for the existence of binary LCPs of codes. As a result, we solve an open problem…
We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+\delta$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that…
A classic result due to Douglas establishes that, for odd spread $k$ and dimension $d=\frac{1}{2}(3k+3)$, all maximum length $(d,k)$ circuit codes are isomorphic. Using a recent result of Byrnes we extend Douglas's theorem to prove that,…
A code is called a $q$-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index $i$ and a received word $w$ close to an encoding of a message $x$, outputs $x_i$ by querying only at most $q$…
We obtain new linear programming (LP) and constructive bounds for the covering radius of binary orthogonal arrays of strength $2k$. Our LP bounds develop in two alternative scenarios. First, if a point $y \in F_2^n$, where the covering…
Polycyclic codes offer a natural generalization of cyclic codes and provide a broader algebraic framework for constructing linear codes with good parameters. In this paper, we study binary polycyclic codes associated with powers of…
It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved this conjecture for…
LCD BCH codes are an important class of cyclic codes which have efficient encoding and decoding algorithms, but their parameters are difficult to determine. The objective of this paper is to study the LCD BCH codes of length…
Let $\mathtt{R}$ be a finite commutative chain ring with the maximal ideal $\gamma\mathtt{R}$ of nilpotency index $e\geq 2,$ and let $\check{\mathtt{R}}=\mathtt{R}/\gamma^{s}\mathtt{R}$ for some positive integer $ s< e.$ In this paper, we…