Codes, Differentially $\delta$ -Uniform Functions, and $t$ -Designs

Boolean functions, coding theory and t-designs have close connections and interesting interplay. A standard approach to constructing t-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting t-designs. However, some linear codes hold t-designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a t-transitive or t-homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new t-designs. To this end, a general theory for punctured and shortened codes of linear codes supporting t-designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially δ-uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system S(2, 4, 2ⁿ) are produced in this paper.