One-weight codes in some classes of group rings
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One‑weight codes in some classes of group rings Raul Antonio Ferraz1 · Ruth Nascimento Ferreira2 Received: 22 July 2019 / Revised: 28 September 2020 / Accepted: 29 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let 𝔽q be a finite field with q elements and G be a finite abelian group. In this work we gave conditions to ensure that a code in 𝔽q G is a one-weight code in the case when G is a cyclic group with n elements, such that gcd(n, q) = 1 , and also when G is an abelian group. Keywords One weight codes · Cyclic group · Abelian group Mathematics Subject Classification 94B05
1 Introduction A type of code of particular interest is the case when the code has constant weight, that is, all its non zero words have the same Hamming weight. Many works in this area consider binary one-weight codes which have many applications, for example in mobile communication. Nowadays, the interest in non binary one-weight codes is also increasing (see for example [4]). k −1) In [11], Vega characterized one-weight codes in 𝔽q Cn , with n = 𝜆(qq−1 , where 𝜆 divides q − 1 , and obtained the number of one-weight codes in this ring. His paper made use of polynomial tools, specially linear recurrence sequences. Inspired by these results, we use group ring theory to extend his results to codes in 𝔽q Cn , with n arbitrary such that gcd(n, q) = 1 , that is, when the group ring is semisimple. We then characterize one-weight codes in abelian group rings, first when the group ring is semisimple and then in the general case. * Raul Antonio Ferraz [email protected] Ruth Nascimento Ferreira [email protected] 1
Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, 05508‑090 São Paulo SP, Brazil
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Universidade Tecnológica Federal do Paraná, Avenida Professora Laura Pacheco Bastos, 800, 85053‑510 Guarapuava PR, Brazil
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R. A. Ferraz, R. N. Ferreira
We recall that, in the context of group rings, a q-ary code is an ideal of a group ring 𝔽q G , and we shall use throughout these words as synonyms. This view point started with the works of Berman [1, 2] and MacWilliams [8] who worked mainly with Abelian groups. Since then, many authors have used group rings to study errorcorrecting codes. See, for example, [6, 7, 9, 10]. Throughout the paper, all codes considered are linear cyclic (or Abelian) over a field 𝔽q are viewed as ideals in appropriate group rings. All the groups considered in this paper are finite.
2 Cyclic groups Let n, q be positive integers, q a power of a prime rational integer such that gcd(n, q) = 1 . Let 𝔽q be the finite field with q elements and G be an abelian group of order n. Then the group ring 𝔽q G is semisimple. Given a subgroup H of G, we denote ̂ the element H ̂ = 1 ∑ h ∈ 𝔽q G which is an idempotent in 𝔽q G . In [3], the by H �H� h∈H
following definition is given:
Definition 1 ( [3, Definition 2.2]) A primitive idempotent e in 𝔽q G is called essential ̂ = 0 , for all subgroups H of G such that H ≠ 1. if eH
� = e} . Let e be
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