Algorithm

Discrete mathematics

Boolean function

Automorphism

Set (abstract data type)

Automorphism group

Computer science

Code (set theory)

Decoding methods

Coding theory

Mathematics

Block code

Combinatorics

Programming language

Steiner system

Combinatorial design

Binary number

Linear code

Arithmetic

IEEE Transactions on Information Theory

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, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) are produced in this paper.

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