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This report is a survey of self-dual binary codes. We present the fundamental MacWilliams identity and Gleason’s theorem on self-dual binary codes. We also examine the upper bound of minimum weights of self-dual binary codes using the extremal weight enumerator formula. We describe the shadow code of a self-dual code and the restrictions of the weight enumerator of the shadow code. Then using the restrictions, we calculate the weight enumerators of self-dual codes of length 38 and 40 and we obtain the known weight enumerators of this lengths. Finally, we investigate the Gaborit-Otmani experimental construction of selfdual binary codes. This construction involves a fixed orthogonal matrix, and we compare the result to the results obtained using other orthogonal matrices. / text

The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation between the weight enumerator of a linear code and the weight enumerator of its dual code. For certain codes C coming from families of varieties where it is not known how to determine the distribution of point counts directly, we analyze low-weight codewords of the dual code and apply the MacWilliams theorem and its generalizations to gain information about the weight enumerator of C. These low-weight dual codes can be described in terms of point sets that fail to impose independent conditions on this family of varieties. Our main results concern rational point count distributions for del Pezzo surfaces of degree 2, and for certain families of genus 1 curves. These weight enumerators have interesting geometric and coding theoretic applications for small q. / Mathematics

In the thesis we study two dimensional simplicial complexes and linear codes. We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex ∆ such that C is a punctured code of the kernel ker ∆ of the incidence matrix of ∆ over F and dim C = dim ker ∆. We call this simplicial complex a geometric representation of C. We show that every linear code C over a primefield is triangular representable. In the case of finite primefields we construct a geometric representation such that the weight enumerator of C is obtained by a simple formula from the weight enumerator of the cycle space of ∆. Thus the geometric representation of C carries its weight enumerator. Our motivation comes from the theory of Pfaffian orientations of graphs which provides a polynomial algorithm for weight enumerator of the cut space of a graph of bounded genus. This algorithm uses geometric properties of an embedding of the graph into an orientable Riemann surface. Viewing the cut space of a graph as a linear code, the graph is thus a useful geometric representation of this linear code. We study embeddability of the geometric representations into Euclidean spaces. We show that every binary linear code has a geometric representation that can be embed- ded into R4 . We characterize...