Binary divisible codes of maximum dimension

摘要

Divisible codes were introduced by H.N. Ward in 1981. A divisible code is a linear code over a finite field whose codewords all have weights divisible by some integer Δ > 1, where Δ is called a divisor of the code. A binary linear code is said to be of (divisibility) level e if e is the greatest integer such that 2~e is a divisor of the code. The doubly-even binary self-dual codes may be viewed as level 2 divisible codes attaining the largest conceivable dimension for their lengths. In this paper, we give an exact upper bound for the dimension of binary divisible codes in terms of code length and divisibility level (when the level is at least 3) and prove the uniqueness up to equivalence of the code attaining this bound, given the hypothesis that a certain non-zero weight exists. We also prove that the hypothesis is true for level 3 divisible codes of maximum dimension with relatively short lengths.