On Bounded Weight Codes

TL;DR

This paper investigates the maximum size of binary codes with constraints on length, minimum distance, and minimum weight, providing asymptotic characterizations and improved bounds using semidefinite programming.

Contribution

It introduces a new function B(N,D,W) for bounded weight codes, characterizes its exponential growth rate, and derives tighter upper bounds via SDP methods.

Findings

Exponential growth rate of B(N,D,W) matches that of A(N,D) or A(N,D,W) depending on weight fraction.

Derived non-asymptotic upper bounds improve upon the second Johnson bound.

Bounds are tight for specific parameter values.

Abstract

The maximum size of a binary code is studied as a function of its length N, minimum distance D, and minimum codeword weight W. This function B(N,D,W) is first characterized in terms of its exponential growth rate in the limit as N tends to infinity for fixed d=D/N and w=W/N. The exponential growth rate of B(N,D,W) is shown to be equal to the exponential growth rate of A(N,D) for w <= 1/2, and equal to the exponential growth rate of A(N,D,W) for 1/2< w <= 1. Second, analytic and numerical upper bounds on B(N,D,W) are derived using the semidefinite programming (SDP) method. These bounds yield a non-asymptotic improvement of the second Johnson bound and are tight for certain values of the parameters.