On Error-Correcting Codes and Invariant Linear Forms

Abstract

Given a code C, invariant linear forms are used to study the designs afforded by codewords of a fixed weight. The most important theorem relating codes and designs is due to Assmus and Mattson [J. Combin. Theory, 6 (1969), pp. 122–151], and this theorem is extended in different ways. For extremal self dual codes over the fields $\mathbb{F}_2 $ and $\mathbb{F}_3 $, it is proved that the t-designs afforded by the codewords of any fixed weight exhibit extra regularity with respect to $( t + 2 )$-sets. The same is true for the design afforded by the codewords of minimum weight in an extremal self-dual code over $\mathbb{F}_4 $. The invariant linear forms are also used to construct Boolean designs with several block sizes, extending previous work by Safavi-Naini and Blake [Utilitas Math., 14 (1978), pp. 49–63], [Ars Combin., 7 (1979), pp. 135–151], [Inform. and Control, 42 (1986), pp. 261–282].