Constant-weight code

In coding theory, a constant-weight code, also called an m of n code, is an error detection and correction code where all codewords share the same Hamming weight. The one-hot code and the balanced code are two widely used kinds of constant-weight code.

The theory is closely connected to that of designs (such as t-designs and Steiner systems). Most of the work on this very vital field of discrete mathematics is concerned with binary constant-weight codes.

Binary constant-weight codes have several applications, including frequency hopping in GSM networks. Most barcodes use a binary constant-weight code to simplify automatically setting the threshold. Most line codes use either a constant-weight code, or a nearly-constant-weight paired disparity code. In addition to use as error correction codes, the large space between code words can also be used in the design of asynchronous circuits such as delay insensitive circuits.

Constant-weight codes, like Berger codes, can detect all unidirectional errors.

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 ${\displaystyle n}$, Hamming distance ${\displaystyle d}$, and weight ${\displaystyle w}$? This number is called ${\displaystyle A(n,d,w)}$.

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