Generalizations of Gleason's theorem on weight enumerators of self-dual codes

Abstract

Gleason has recently shown that the weight enumerators of binary and ternary self-dual codes are polynomials in two given polynomials. In this paper it is shown that classical invariant theory permits a straightforward and systematic proof of Gleason's theorems and their generalizations. The joint weight enumerator of two codes (analogous to the joint density function of two random variables) is defined and shown to satisfy a MacWilliams theorem. Invariant theory is then applied to generalize Gleason's theorem to the complete weight enumerator of self-dual codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(3)</tex> , the Lee metric enumerator over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(5)</tex> (given by Klein in 1884!) and over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(7)</tex> (given by Maschke in 1893!), the Hamming enumerator over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q)</tex> , and over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(4)</tex> with all weights divisible by 2, the joint enumerator of two self-dual codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(2)</tex> , and a number of other results.