Combinatorics of MacWilliams identities
Title: Combinatorics of MacWilliams identities
Speaker: Alberto Ravagnani (University of Neuchatel)
Date: Monday, 19th October 2015
Time: 4pm – 5pm
Location: Agriculture. 1.01 (Seminar Room).
Abstract:
In coding theory, a MacWilliams identity expresses linear relations between the weight distribution of a code and the weight distribution of the dual code. The transformation is described by certain numbers called Krawtchouk coefficients. When studying additive codes in finite abelian groups, the code and the dual code are subsets of different ambient spaces, and their weight enumerators refer in general to different weight functions. Invertible MacWilliams identities hold when the weights are mutually compatible. A major problem in this area is the construction of mutually compatible weights, and the computation of the associated Krawtchouk coefficients.Using a combinatorial method, we construct a family of mutually compatible weight functions on finite abelian groups that yield invertible MacWilliams identities. The weights are obtained composing a support map with the rank function of a graded lattice with certain regularity properties. We express the corresponding Krawtchouk coefficients in terms of the combinatorial invariants of the underlying lattice, giving a method to compute them.The most important weight functions studied in coding theory belong, up to equivalence, to the class of weight functions that we introduce. This allows in particular to compute some classical Krawtchouk coefficients with combinatorial techniques.Finally, we show applications of our approach to some enumerative problems over finite fields.
Series: Algebra and Number Theory
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