A geometric approach to classifying Griesmer codes

Abstract

We develop a geometric framework for the determination of codes meeting the Griesmer bound that involves a classification of ((q + 1)x, x) generalized minihypers in PG(2, q). These are multisets: each point of the plane has a multiplicity, and the strength of a line is the sum of its point multiplicities. The minihyper requirement is that each line has strength at least x, with some line having strength exactly x. When $1\leq x\leq q-1$ , the code corresponding to the minihyper meets the Griesmer bound. A crucial observation is that from any ((q + 1)x, x)-minihyper ${\mathcal{M}}$ we can obtain a ((q + 1)(x + 1), x + 1)-minihyper ${\mathcal{M}}^{\prime }$ by selecting a line and increasing its point multiplicities by 1. Minihyper ${\mathcal{M}}^{\prime }$ is a “child” of its “parent” ${\mathcal{M}}$ , and classification focuses on minihypers that are “orphans” with no parent. This inductive set-up expedites the search for Griesmer codes. There is a divisibility result parallel to one for codes meeting the Griesmer bound, but somewhat stronger. Divisibility simplifies the determination of orphans, and geometric objects such as ovals and blocking sets often occur in the descriptions. For q a prime, there are no nontrivial orphans, and we describe the non-trivial orphans for q = 4, 8 and 9 completely, along with their descendants.