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- W4313479007 abstract "A pair (n,q) with q a prime power and n a positive integer for which gcd(n,q)=1 is non-standard if one of the following properties holds, which we will show to be equivalent. − The group Un,q of n-th roots-of-unity in the splitting field Fqm of xn−1 over Fq is fixed set-wise by an Fq-linear map on Fqm that is not of the form x⟶αxqj. − There is a non-cyclic linear recurring sequence s of period n (so with s not of the form si=αξi for i≥0) with associated characteristic polynomial being irreducible over Fq, such that Un,q={s0,…,sn−1}. In that case, the group Un,q is called non-standard by Brison and Nogueira, who studied this phenomenon in a sequence of papers. − A q-ary irreducible cyclic code of length n has “extra” permutation automorphisms (some well-known examples are the (duals of the) Golay codes and binary simplex codes for n≥7). We will refer to such codes as non-standard irreducible cyclic codes or NSIC-codes. We first investigate non-standard linear recurring sequence subgroups and establish the equivalence between the first and second of the above properties. Then we investigate permutation automorphisms of general (repeated-root) cyclic codes and irreducible cyclic codes and establish the equivalence between the first and third of these properties. We also introduce a notion of non-standardness for general cyclic codes, and relate it to our earlier definition of NSIC-codes. This paper is the first part of an expanded version of the arXiv paper [28]." @default.
- W4313479007 created "2023-01-06" @default.
- W4313479007 creator A5089300685 @default.
- W4313479007 date "2023-02-01" @default.
- W4313479007 modified "2023-09-27" @default.
- W4313479007 title "Non-standard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes I" @default.
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- W4313479007 doi "https://doi.org/10.1016/j.ffa.2022.102146" @default.
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