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Special functions, coding theory and $t$-designs have close connections andinteresting interplay. A standard approach to constructing $t$-designs is theuse of linear codes with certain regularity. The Assmus-Mattson Theorem and theautomorphism groups are two ways for proving that a code has sufficientregularity for supporting $t$-designs. However, some linear codes hold$t$-designs, although they do not satisfy the conditions in the Assmus-MattsonTheorem and do not admit a $t$-transitive or $t$-homogeneous group as asubgroup of their automorphisms. The major objective of this paper is todevelop a theory for explaining such codes and obtaining such new codes andhence new $t$-designs. To this end, a general theory for punctured andshortened codes of linear codes supporting $t$-designs is established, ageneralized Assmus-Mattson theorem is developed, and a link between $2$-designsand differentially $\delta$-uniform functions and $2$-designs is built. Withthese general results, binary codes with new parameters and known weightdistributions are obtained, new $2$-designs and Steiner system $S(2, 4, 2^n)$are produced in this paper.

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