MacWilliams Identities for $m$-tuple Weight Enumerators

Abstract

Since MacWilliams proved the original identity relating the Hamming weight enumerator of a linear code to the weight enumerator of its dual code there have been many different generalizations, leading to the development of $m$-tuple support enumerators. We prove a generalization of theorems of Britz and of Ray-Chaudhuri and Siap, which build on earlier work of Kløve, Shiromoto, Wan, and others. We then give illustrations of these $m$-tuple weight enumerators.

arXiv comment17 pages. Jesse Geneson discovered a typo in the statement of Corollary 13. The expression inside the parentheses on the right side of the equation should read, "(X+(q^{2m}-1)Y,X-Y)" instead of "(X+(q^m-1)Y,X-Y)". The published version of the article also contains this typo