Projective codes meeting the Griesmer bound

AbstractWe present a brief survey of projective codes meeting the Griesmer bound. Methods for constructing large families of codes as well as sporadic codes meeting the bound are given. Current research on the classification of codes meeting the Griesmer bound is also presented

On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)

We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound

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