# Codes, differentially $\delta$-uniform functions and $t$-designs

## TL;DR

This paper develops a new theoretical framework linking linear codes, $t$-designs, and differentially $oldsymbol{ extit{ extdelta}}$-uniform functions, leading to the construction of new codes and combinatorial designs beyond classical criteria.

## Contribution

It introduces a generalized theory for punctured and shortened codes, extends the Assmus-Mattson theorem, and connects $2$-designs with differentially $ extdelta$-uniform functions, enabling new code and design constructions.

## Key findings

- Constructed binary codes with new parameters and known weight distributions.
- Produced new 2-designs and Steiner systems S(2, 4, 2^n).
- Extended the theoretical understanding of codes supporting $t$-designs.

## Abstract

Special 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 $\delta$-uniform functions and $2$-designs is built. With these general results, binary codes with new parameters and known weight distributions are obtained, new $2$-designs and Steiner system $S(2, 4, 2^n)$ are produced in this paper.