| 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 Rn to linear codes C ⊆ Fn2 [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 ⊆ Fn2 [7], and an analytic proof of the Lee weight MacWilliams Identity exists for self-orthogonal, C ⊆ C ⊥, linear codes C ⊆ Fnp [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 ⊆ Matm xn( F2 ). (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 ⊆ Mat2x n( F2 ) and an analytic proof of the rank weight MacWilliams Identity for linear codes C ⊆ Mat2x2( F2 ). 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 ⊆ Fnp . 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. |
