Definitions

# Constant-weight code

In coding theory, a constant-weight code is an error detection and correction code where all codewords share the same Hamming weight. The theory is closely connected to that of designs (such as t-designs and Steiner systems) and has several applications, including frequency hopping in GSM networks. Most of the work on this very vital field of discrete mathematics is concerned with binary constant-weight codes.

## A(n,d,w)

The central problem regarding constant-weight codes is the following: what is the maximum number of codewords in a binary constant-weight code with length $n$, Hamming distance $d$, and weight $w$? This number is called $A\left(n,d,w\right)$.

Apart from some trivial observations, it is generally impossible to compute these numbers in a straightforward way. Upper bounds are given by several important theorems such as the first and second Johnson bounds, and better upper bounds can sometimes be found in other ways. Lower bounds are most often found by exhibiting specific codes, either with use of a variety of methods from discrete mathematics, or through heavy computer searching. A large table of such record-breaking codes was published in 1990, and an extension to longer codes (but only for those values of $d$ and $w$ which are relevant for the GSM application) was published in 2006.