The projective general linear group $${\mathrm {PGL}}(2,2^m)$$ and linear codes of length $$2^m+1$$
| Autor: | Cunsheng Ding, Vladimir D. Tonchev, Chunming Tang |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Applied Mathematics
Dimension (graph theory) 020206 networking & telecommunications 0102 computer and information sciences 02 engineering and technology Permutation group 01 natural sciences Linear code Upper and lower bounds Computer Science Applications Combinatorics 010201 computation theory & mathematics Cyclic code Projective line 0202 electrical engineering electronic engineering information engineering Projective linear group Invariant (mathematics) Mathematics |
| Zdroj: | Designs, Codes and Cryptography. 89:1713-1734 |
| ISSN: | 1573-7586 0925-1022 |
| DOI: | 10.1007/s10623-021-00888-2 |
| Popis: | Let $$q=2^m$$ . The projective general linear group $${\mathrm {PGL}}(2,q)$$ acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $${\mathrm {GF}}(2^h)$$ that are invariant under $${\mathrm {PGL}}(2,q)$$ are trivial codes: the repetition code, the whole space $${\mathrm {GF}}(2^h)^{2^m+1}$$ , and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all $$3-(q+1,k,\lambda )$$ designs that are invariant under $${\mathrm {PGL}}(2,q)$$ are determined. The second objective is to present two infinite families of cyclic codes over $${\mathrm {GF}}(2^m)$$ such that the set of the supports of all codewords of any fixed nonzero weight is invariant under $${\mathrm {PGL}}(2,q)$$ , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $$[q+1,q-3,4]_q$$ , where $$q=2^m$$ , and $$m\ge 4$$ is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3- $$(q+1,4,2)$$ design. A code from the second family has parameters $$[q+1,4,q-4]_q$$ , $$q=2^m$$ , $$m\ge 4$$ even, and the minimum weight codewords support a 3- $$(q +1,q-4,(q-4)(q-5)(q-6)/60)$$ design, whose complementary 3- $$(q +1, 5, 1)$$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $${\mathrm {GF}}(q)$$ that can support a 3- $$(q +1,q-4,(q-4)(q-5)(q-6)/60)$$ design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound. |
| Databáze: | OpenAIRE |
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