Title: Kerdock codes and around Author: Kateřina Teplá Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc., Department of algebra Abstract: Kerdock codes form a family of nonlinear codes, that contains more codewords than any known linear code with the same parameters. The main goal of this thesis is a connection of Kerdock codes with other areas of mathematics, mainly orthogonal geometry, combinatorics and cryptogra- phy. It describes theory of symplectic and quadratic forms on vector spaces of characteristic 2 and its relationship to Kerdock codes. Then it is pro- ven, that codewords of Kerdock code of constant weight form combinatorial 3-design. Finally usage of Kerdock codes in construction of Boolean bent functions and t-resilient functions, that are basis of many cryptographic pri- mitives, is analysed. Keywords: Kerdock code, Kerdock set, t-design, resilient function 1

A t-design is a combinatorial structure consisting of a collection of blocks over a set of points satisfying certain properties. The existence of t-designs given a set of parameters can be reduced to finding nonnegative integer solutions to a given integer matrix equation. The matrix in this equation can be quite large, but by prescribing the automorphism group of the design, the matrix in the equation can be made more manageable so as to allow the equation to be solved via integer programming tools; this fact was developed by Kramer and Mesner. Algorithms to generate the matrix equation generally follow a simple template. In this thesis, a generic framework for generating the Kramer-Mesner matrix equation and solving it via integer programming is presented.

A t-design is a combinatorial structure consisting of a collection of blocks over a set of points satisfying certain properties. The existence of t-designs given a set of parameters can be reduced to finding nonnegative integer solutions to a given integer matrix equation. The matrix in this equation can be quite large, but by prescribing the automorphism group of the design, the matrix in the equation can be made more manageable so as to allow the equation to be solved via integer programming tools; this fact was developed by Kramer and Mesner. Algorithms to generate the matrix equation generally follow a simple template. In this thesis, a generic framework for generating the Kramer-Mesner matrix equation and solving it via integer programming is presented.