Optimal binary linear codes of dimension at most seven

Abstract

We classify optimal $\text{[n,k,d]}$ binary linear codes of dimension $\text{⩽7}$, with one exception, where by optimal we mean that no $\text{[n−1,k,d],}\text{[n+1,k+1,d]}$, or $\text{[n+1,k,d+1]}$ code exists. In particular, we present (new) classification results for codes with parameters [40,7,18], [43,7,20], [59,7,28], [75,7,36], [79,7,38], [82,7,40], [87,7,42], and [90,7,44]. These classifications are accomplished with the aid of the first author's computer program Extension for extending from residual codes, and the second author's program Split.