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NONBINARY DELSARTE–GOETHALS CODES AND FINITE SEMIFIELDS

Part of: Finite fields and commutative rings (number-theoretic aspects) Other nonassociative rings and algebras Forms and linear algebraic groups

Published online by Cambridge University Press:  07 May 2020

IGNACIO F. RÚA Open the ORCID record for IGNACIO F. RÚA [Opens in a new window]
IGNACIO F. RÚA*
Affiliation:
Departamento de Matemáticas, Universidad de Oviedo, Oviedo, Spain, e-mail: rua@uniovi.es
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Abstract

Symplectic finite semifields can be used to construct nonlinear binary codes of Kerdock type (i.e., with the same parameters of the Kerdock codes, a subclass of Delsarte–Goethals codes). In this paper, we introduce nonbinary Delsarte–Goethals codes of parameters $(q^{m+1}\ ,\ q^{m(r+2)+2}\ ,\ {\frac{q-1}{q}(q^{m+1}-q^{\frac{m+1}{2}+r})})$ over a Galois field of order $q=2^l$ , for all $0\le r\le\frac{m-1}{2}$ , with m ≥ 3 odd, and show the connection of this construction to finite semifields.

MSC classification

Primary: 11T71: Algebraic coding theory; cryptography 11E08: Quadratic forms over local rings and fields
Secondary: 17D99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Special Issue S1: Workshop on Nonassociative algebras and their applications , December 2020 , pp. S186 - S205
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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