Related papers: On binary codes with distances $d$ and $d+2$
A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $\alpha_d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common…
The Johnson-type upper bound on the maximum size of a code of length $n$, distance $d=2w-1$ and constant composition ${\overline{w}}$ is $\lfloor\dfrac{n}{w_1}\rfloor$, where $w$ is the total weight and $w_1$ is the largest component of…
Line systems passing through the origin of the $d$ dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least $2(d-1)(d-2)$, and this…
We study the maximum length of $q$-ary codes as a function of alphabet size, code size, and Singleton defect. For an $(n, M, d)_q$ code with dimension $\kappa = \log_q M \ge 2$ and Singleton defect $s = n - \lceil\kappa\rceil + 1 - d$, we…
We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a~function of the gap to…
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,…
Given a binary nonlinear code, we provide a deterministic algorithm to compute its weight and distance distribution, and in particular its minimum weight and its minimum distance, which takes advantage of fast Fourier techniques. This…
Codes for storage systems aim to minimize the repair locality, which is the number of disks (or nodes) that participate in the repair of a single failed disk. Simultaneously, the code must sustain a high rate, operate on a small finite…
This paper starts by considering the minimization of the Renyi divergence subject to a constraint on the total variation distance. Based on the solution of this optimization problem, the exact locus of the points $\bigl( D(Q\|P_1),…
It was shown in \cite{GXY18} that the length $n$ of a $q$-ary linear locally recoverable code with distance $d\ge 5$ is upper bounded by $O(dq^3)$. Thus, it is a challenging problem to construct $q$-ary locally recoverable codes with…
We consider weighted Reed-Muller codes over point ensemble $S_1 \times...\times S_m$ where $S_i$ needs not be of the same size as $S_j$. For $m = 2$ we determine optimal weights and analyze in detail what is the impact of the ratio…
Recently, some infinite families of binary minimal and optimal linear codes are constructed from simplicial complexes by Hyun {\em et al}. Inspired by their work, we present two new constructions of codes over the ring $\Bbb F_2+u\Bbb F_2$…
Let $d_1\leq d_2\leq\ldots\leq d_{n\choose 2}$ denote the distances determined by $n$ points in the plane. It is shown that $\min\sum_i (d_{i+1}-d_i)^2=O(n^{-6/7})$, where the minimum is taken over all point sets with minimal distance $d_1…
We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length $n$ of all QMDS codes with local dimension $D$ and distance $d \geq 3$ is bounded by $n \leq D^2 + d - 2$. We obtain their weight…
Binary constant weight codes have important applications and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose a new construction of…
In this work, we study two types of constraints on two-dimensional binary arrays. In particular, given $p,\epsilon>0$, we study (i) The $p$-bounded constraint: a binary vector of size $m$ is said to be $p$-bounded if its weight is at most…
The classical way of extending an $[n, k, d]$ linear code $\C$ is to add an overall parity-check coordinate to each codeword of the linear code $\C$. This extended code, denoted by $\overline{\C}(-\bone)$ and called the standardly extended…
We derive theoretical upper and lower bounds on the maximum size of DNA codes of length n with constant GC-content w and minimum Hamming distance d, both with and without the additional constraint that the minimum Hamming distance between…
$A'(n,d,e)$, the smallest $\ell$ for which every binary error-correcting code of length $n$ and minimum distance $d$ is decodable with a list of size $\ell$ up to radius $e$, is determined for all $d\geq 2e-3$. As a result, $A'(n,d,e)$ is…
Dihedral codes, particular cases of quasi-cyclic codes, have a nice algebraic structure which allows to store them efficiently. In this paper, we investigate it and prove some lower bounds on their dimension and minimum distance, in analogy…