Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum size of a binary code of length $n$ and minimum distance $d$. The well-known Gilbert-Varshamov bound asserts that $A_2(n,d) \geq 2^n/V(n,d-1)$, where $V(n,d) = \sum_{i=0}^{d} {n \choose i}$ is the volume of a Hamming sphere of radius $d$. We show that, in fact, there exists a positive constant $c$ such that $$ A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1) $$ whenever $d/n \le 0.499$. The result follows by recasting the Gilbert- Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed.
We show that every self-orthogonal code over $\mathbb F_q$ of length $n$ can be extended to a self-dual code, if there exists self-dual codes of length $n$. Using a family of Galois towers of algebraic function fields we show that over any nonprime field $\mathbb F_q$, with $q\geq 64$, except possibly $q=125$, there are self-dual codes better than the asymptotic Gilbert--Varshamov bound.
We propose and prove an existential theorem for entanglement-assisted asymmetric quantum error correction. Then we demonstrate its superiority over the conventional one.
Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \geq n!/V(n,d-1)$ where $V(n,d)$ is the volume of a Hamming sphere of radius $d$ in $§_n$. Recently, Gao, Yang, and Ge showed that this bound can be improved by a factor $Ω(\log n)$, when $d$ is fixed and $n \to \infty$. Herein, we consider the situation where the ratio $d/n$ is fixed and improve the Gilbert-Varshamov bound by a factor that is \emph{linear in $n$}. That is, we show that if $d/n < 0.5$, t…
Hopping cyclic codes (HCCs) are (non-linear) cyclic codes with the additional property that the $n$ cyclic shifts of every given codeword are all distinct, where $n$ is the code length. Constant weight binary hopping cyclic codes are also known as optical orthogonal codes (OOCs). HCCs and OOCs have various practical applications and have been studied extensively over the years. The main concern of this paper is to present improved Gilbert-Varshamov type lower bounds for these codes, when the minimum distance is bounded below by a linear factor of the code length. For HCCs, we improve the previ…
We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming, Johnson spaces and symmetric groups decay exponentially as their centers drift apart. To verify condition (iii), we prove some large deviation inequalities `on a slice' for functions with Lipschitz conditions. We then use these estimates on intersecti…
In the first part of the paper we consider binary input channels that are not necessarily stationary and show how positive rates can be achieved using codes constrained to be within predetermined dictionaries. We use a Gilbert-Varshamov-like argument to obtain the desired rate achieving codes. Next we study the corresponding problem for channels with arbitrary alphabets and use conflict-set decoding to show that if the dictionaries are contained within nice sets, then positive rates are achievable.
In this paper, we study the redundancy of linear codes with graph constraints. First we consider linear parity check codes based on bipartite graphs with diversity and with generalized graph constraints. We describe sufficient conditions on the constraint probabilities and use the probabilistic method to obtain linear codes that achieve the Gilbert-Varshamov redundancy bound in addition to satisfying the constraints and the diversity index. In the second part we consider a generalization of graph capacity which we call as the fractional graph capacity and use the probabilistic method to determ…
This paper gives a brief survey of binary single-deletion-correcting codes. The Varshamov-Tenengolts codes appear to be optimal, but many interesting unsolved problems remain. The connections with shift-register sequences also remain somewhat mysterious.
Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall's $τ$-metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers $n$ and $d$, let $\mathcal{C}_{B}(n,d)$ denote the maximum size of a pe…
The capacity of a graph is defined as the rate of exponential grow of independent sets in the strong powers of the graph. In strong power, an edge connects two sequences if at each position letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed among sequences which differ in more than a fraction $δ$ of coordinates. For odd cycles, we derive an upper bound on the corresponding rate which combines Lovász' bound on the capacity with Delsarte's linear programming bounds on the minimum distance of codes in Hamming spaces. For the pentagon,…
We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a codeword. We demonstrate that the number of such balls that cover the entire code alphabet cannot exceed this maximum size. Our approach can be applied to codes $i)$ with elements over arbitrary code alphabets, and $ii)$ under a broad class of distance measures, thereby ensuring the generality of our formula. Our formula indicates that the maximum code size…
A longstanding open problem in coding theory is to determine the best (asymptotic) rate $R_2(δ)$ of binary codes with minimum constant (relative) distance $δ$. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte's linear programs. To date these results remain the best known lower and upper bounds on $R_2(δ)$ with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the bloc…
