On the size of maximal binary codes with 2, 3, and 4 distances

TL;DR

This paper determines the exact maximum sizes of binary codes with two distances for all lengths greater than or equal to 6, and for certain constant weight codes with 2, 3, and 4 distances, expanding known bounds.

Contribution

It provides exact sizes of maximal binary codes with two distances for all lengths ≥6 and for specific constant weight codes with 2, 3, and 4 distances, improving previous bounds.

Findings

Exact sizes for binary codes with two distances for all lengths ≥6

Exact sizes for constant weight codes with 2, 3, and 4 distances for certain parameters

Extended the known bounds and determined precise maximum code sizes

Abstract

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths $n\ge 6$ as well as the exact size of maximal binary constant weight codes with 2,3, and 4 distances for several values of the weight and for all but small lengths.