Optimal binary linear codes of dimension at most seven

AbstractWe classify optimal [n,k,d] binary linear codes of dimension ⩽7, with one exception, where by optimal we mean that no [n−1,k,d],[n+1,k+1,d], or [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

A Characterization of Some Minihypers in a Finite Projective Geometry PG(t, 4)

Recently, Hamada and Deza [8] gave a complete characterization of all {vα + 1 + vβ + 1 + vγ + 1, vα + vβ + vγ; t, q}-minihypers for any integers α, β, γ, t and any prime power q such that q ⩾ 5 and either 0 ⩽ α = β < γ < t or 0 ⩽ α < β = γ < t where vl = (ql− 1)/(q − 1) for any integer l ⩾ 0. The purpose of this paper is to characterize all {vα + 1 + vβ + 1 + vγ + 1, vα + vβ + vγ; t, q}-minihypers for any integers t, q, α, β and γ such that q = 4 and either (a) 0 ⩽ α < β = γ < t or (b) 0 ⩽ α = β < γ < t and γ ≠ α + 1. Using those results, all (n, k, d ; 4)-codes meeting the Griesmer bound are characterized for the case k ⩾ 3 and d = 4k−1 − 4α − 4β − 4γ

An improvement of the Griesmer bound for some small minimum distances

AbstractIn this paper we give some lower and upper bounds for the smallest length n(k, d) of a binary linear code with dimension k and minimum distance d. The lower bounds improve the known ones for small d. In the last section we summarize what we know about n(8, d)