Two families of two-weight codes over $$\mathbb {Z}_4$$ Z 4
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Two families of two-weight codes over Z4 Minjia Shi1
· Wang Xuan1 · Patrick Solé2
Received: 21 March 2020 / Revised: 1 July 2020 / Accepted: 20 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Two infinite families of Z4 -codes with two nonzero Lee weights are constructed by their generator matrices. Their Gray images are nonlinear with the same weight distribution as that of the two-weight binary codes of type SU1 in the sense of (Calderbank, Kantor, 1986). Keywords Two-weight codes · Projective codes · Strongly regular graphs Mathematics Subject Classification 94B 05 · 05E 30
1 Introduction An organic connection between two-weight codes over fields and strongly regular graphs (SRGs) was discovered in the 1970s [3], and is well-documented in [1]. A partial census of known constructions based on the arithmetic of finite fields appeared in the classic paper [2]. More recently, several new techniques of construction involve trace codes over different rings, and Gray maps [9–13,15]. Especially, in [14], the authors focused on the construction of one-weight and two-weight codes over Z4 . Later, in [16], the authors considered the linearity of the constructed codes. A natural question then, is whether there are other kinds of constructions of two-weight codes over Z4 . Motivated by the works listed above, in the present paper, we revisit two-weight codes over Z4 and their linearity. This alphabet has been on the forefront of research in the domain of codes over rings since the prize awarded paper [4]. More background material can be found in the recent book [7]. We give an infinite family of two-weight projective codes over Z4 by their explicit generator matrices. Their Gray images are proved to be nonlinear with the same weight distribution as the two-weight binary codes of type SU1 from [2]. The coset graphs of the dual codes are shown to be SRGs, and determined completely. Thus they produce the
Communicated by P. Charpin.
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Minjia Shi [email protected]
1
Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of Mathematical Sciences, Anhui University, Anhui 230601, China
2
I2M,(Aix-Marseille Univ., Centrale Marseille, CNRS), Marseille, France
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M. Shi et al.
same SRG’s as those given by the SU1 codes, but described as Cayley graphs on a different abelian group. This note is organized in the following way. The next section contains notations and definitions. Section 3 is dedicated to preliminary material. Section 4 derives the main construction. Section 5 studies the linearity of the Gray images of our codes. Section 6 determines the SRGs attached to the Z4 codes of Sect. 4. Section 7 concludes the note.
2 Notations and definitions 2.1 Z4 codes We first recall necessary notations and definitions about codes over Z4 in [17] and [7]. Let Z4 denote the ring of integers modulo four Z4 = {0, 1, 2, 3}. A linear code C over Z4 of length n is a Z4 -submodule of Zn4 . The Lee weights of 0, 1, 2, 3 ∈ Z4 are 0, 1, 2, 1 respectively.
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