## View abstract

### Session S26 - Finite fields and applications

Tuesday, July 20, 17:00 ~ 17:50 UTC-3

## Weights on $\mathbb{Z}/m\mathbb{Z}$ and the MacWilliams Identities

### Jay A. Wood

#### Western Michigan University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc3e80f3bd8f1d6bff9eb00f824ca8667').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc3e80f3bd8f1d6bff9eb00f824ca8667 = 'j&#97;y.w&#111;&#111;d' + '&#64;'; addyc3e80f3bd8f1d6bff9eb00f824ca8667 = addyc3e80f3bd8f1d6bff9eb00f824ca8667 + 'wm&#105;ch' + '&#46;' + '&#101;d&#117;'; var addy_textc3e80f3bd8f1d6bff9eb00f824ca8667 = 'j&#97;y.w&#111;&#111;d' + '&#64;' + 'wm&#105;ch' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakc3e80f3bd8f1d6bff9eb00f824ca8667').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc3e80f3bd8f1d6bff9eb00f824ca8667 + '\'>'+addy_textc3e80f3bd8f1d6bff9eb00f824ca8667+'<\/a>';

There are four well-known weights defined on $\mathbb{Z}/m\mathbb{Z}$: the Hamming weight, the Lee weight, the Euclidean weight, and the homogeneous weight. Each weight determines a weight enumerator for linear codes, counting the number of codewords whose weight is a given value. The MacWilliams identities give a relation between the Hamming weight enumerator of a linear code and that of its dual code. In contrast, the MacWilliams identities tend to fail for the other three weights. We will summarize what is known, with an emphasis on the homogeneous weight, where the MacWilliams identities fail for composite $m > 5$.