Characterization and Classification of Binary Linear Codes With Various Hull Dimensions From an Improved Mass Formula
Shitao Li, Minjia Shi
Abstract
The hull of a linear code over finite fileds is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main purpose of this paper is to obtain the closed mass formula for binary linear codes with various hull dimensions, which simplifies the mass formula obtained by Sendrier in (SIAM J. Discrete Math., 10(2): 282-293, 1997). We show that almost all binary linear codes with ℓ-dimensional hull are odd-like codes with odd-like duals for fixed ℓ. We also study the largest minimum distance of a binary linear [n, k] code with ℓ-dimensional hull. Most importantly, we give a complete classification of binary linear codes with various hull dimensions for n ≤ 12 using a building-up construction, which is confirmed by double-checking with our mass formula. We also give the classification of optimal binary linear [n, k] codes with various hull dimensions for n ≤ 13. Combining with known results, we obtain the classification of (optimal) binary linear codes with small parameters.