Overlap and Covering Polynomials with Applications to Designs and Self-Dual Codes

Abstract

For a binary linear code ${\cal C}$ and a vector u, we define and study the overlap polynomial $F^{(u)}_{{\cal C}}(Z,X,D)$ which is a sum over the words in ${\cal C}$ of terms that takes into account the weight of the element of ${\cal C}$ and its overlap with the vector $u$. We obtain a generalized MacWilliams relation between $F^{(u)}_{{\cal C}}(Z,X,D)$ and $F^{(u)}_{{\cal C}^{\perp}}(Z,X,D)$ and a recursion equation for determining the overlap polynomials of a self-dual code in terms of overlap polynomials corresponding to vectors of lower weight. We apply the recursion to determine the weight enumerators of the cosets of the (putative) [72,36,16] self-dual code of Type II, up to a small number of parameters. The overlap polynomials are expressed in terms of covering polynomials $g^{(u)}_{{\cal C}}(Z,X)$ that are sums over the elements of ${\cal C}$ that cover $u$. The design strength $\delta({\cal C})$ is closely related to the covering polynomials and we use them to get a sharpened version of the Assmus--Mattson theorem. In many cases we are able to conclude that certain codes hold $\delta({\cal C})$-designs but no weight class holds a $1+\delta({\cal C})$-design. We study the design strength of self-dual, doubly even extremal codes and, in the case of length divisible by 24, improve the well-known estimate $\delta({\cal C})\geq 5$ to state that either $\delta({\cal C})\geq 7$ or $\delta({\cal C})=5$ and no weight class holds a 6-design; similar results hold in the case of other lengths. Examples are given using a self-dual code [32,16,8], certain cyclic subcodes of the punctured Reed--Muller code ${\cal R}(2,m)^*$, the first order RM-code, and a [16,9,4] code in which all six of the weight classes hold 2-designs but one nontrivial weight class also holds a 3-design.