Related papers: Almost perfect linear Lee codes of packing radius …
We classify all linear completely regular codes which have covering radius $\rho = 2$ and whose dual are antipodal. For this, we firstly show several properties of such dual codes, which are two-weight codes.
A $\lambda$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $\lambda$ times. The…
The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank…
A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp}$ satisfies $C \cap C^{\perp}$= $\left\{ \mathbf{0}\right\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of…
A binary code of blocklength $n$ and codebook size $M$ is called an $(n,M)$ code, which is studied for memoryless binary symmetric channels (BSCs) with the maximum likelihood (ML) decoding. For any $n \geq 2$, some optimal codes among the…
This paper presents a new explicit infinite family of 2-quasi-perfect $p$-ary Lee codes of length $\frac{q-1}{2}$ and dimension $\frac{q-1}{2}-2k$ for $q = p^k \ge 14$, $p\geq 5$ a prime. Our codes are derived from the generating set $H_q =…
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…
For every $n = 2^k > 8$ there exist exactly $[(k+1)/2]$ mutually nonequivalent $Z_4$-linear extended perfect codes with distance 4. All these codes have different ranks.
It is proven that for any numbers n=2^m-1, m >= 4 and r, such that n - log(n+1)<= r <= n excluding n = r = 63, n = 127, r in {126,127} and n = r = 2047 there exists a propelinear perfect binary code of length n and rank r.
A longstanding open problem in coding theory is to determine the best (asymptotic) rate $R_2(\delta)$ of binary codes with minimum constant (relative) distance $\delta$. An existential lower bound was given by Gilbert and Varshamov in the…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric.…
We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems. Specifically, we prove that if almost all codes (in the class…
This paper shows that there exist Reed--Solomon (RS) codes, over \black{exponentially} large finite fields \black{in the code length}, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving the…
One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the…
In this paper, we study one-Lee weight and two-Lee weight codes over $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, where $u^{2}=0$. Some properties of one-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are given, and a complete…
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…
Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over $\mathbb{F}_q$ having large minimum weights for $q \in \{2,3\}$. Using the characterization, we…
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…
We establish a connection between linear complementary dual (LCD) codes and caps in projective space. Using this framework and the structure theory of maximal caps, we derive nonexistence theorems for LCD codes with minimum distance at…
We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p,$ where $u^2=0,v^2=0,uv=vu.$ These codes are defined as trace codes. They have the…