Analytic Proofs of Certain MacWilliams Identities
David Parker Keyes · CU Scholar (University of Colorado Boulder) · 2011
The relationship between vector codes and lattices has been studied extensively over the past 40 years.Since Leech and Sloane described how to attach lattices in R n to linear codes C ⊆ F n 2 [35], lattices have been attached to codes defined over a variety of finite rings.Much research has been conducted on theta functions defined over these code lattices and their modular properties.Codes C are modeled mathematically as a subset of matrices (codewords) with entries in a finite alphabet B. Weight functions measure the "size" of elements v ∈ C. A weight enumerator is a generating function that encodes the weight distribution of a code.If the code C is a vector space, then its dual C ⊥ is the orthogonal vector space under the dot product.Duality theory for codes was pioneered by Sloane, MacWilliams, and Delsarte [43], [21], [32].MacWilliams Identities are at the center of this theory.MacWilliams Identities are functional equations that relate the weight enumerator of a code to that of its dual.An analytic proof of the Hamming weight MacWilliams Identity exists for linear codes C ⊆ F n 2 [7], and an analytic proof of the Lee weight MacWilliams Identity exists for self-orthogonal, C ⊆ C ⊥ , linear codes C ⊆ F n p [13].We extend the class of codes for which there exists an analytic proof of the MacWilliams Identity.In Chapter 3, we describe how to attach theta functions to matrix codes C ⊆ Mat m×n (F 2 ).(We believe this is the first time theta functions have been attached to matrix codes.)We provide an analytic proof of the column distance weight MacWilliams Identity for linear codes C ⊆ Mat 2×n (F 2 ) and an analytic proof of the rank weight MacWilliams Identity for linear codes C ⊆ Mat 2×2 (F 2 ).In Chapter 4, we improve upon the work of van der Geer, Hirzebruch, Choie, and Jeong [27], [13].We provide an analytic proof of the Hamming weight MacWilliams Identity for linear vector codes C ⊆ F n p .In Chapter 5, we provide a general framework for this problem and study the relationship between codes and theta functions in the context of association schemes.