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- W2964502204 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." @default.
- W2964502204 created "2019-08-13" @default.
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- W2964502204 date "2019-07-30" @default.
- W2964502204 modified "2023-10-01" @default.
- W2964502204 title "Codes, differentially $delta$-uniform functions and $t$-designs" @default.
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- W2964502204 doi "https://doi.org/10.48550/arxiv.1907.13036" @default.
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