Construction of isodual codes from polycirculant matrices
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Construction of isodual codes from polycirculant matrices Minjia Shi1
· Li Xu1 · Patrick Solé2
Received: 30 November 2019 / Revised: 15 July 2020 / Accepted: 25 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over F2 in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over F2 , we show that binary double polycirculant codes are asymptotically good. Keywords Quasi-polycyclic codes · Isodual codes · Formally self-dual codes · Double circulant codes · Trinomials Mathematics Subject Classification Primary 94B05 · Secondary 11C08
1 Introduction Self-dual codes is one of the most fascinating class of codes as witnessed by their many connections with modular forms [25], invariant theory and combinatorial designs [21]. This class has been enlarged, in recent years, to isodual codes that is to say codes that are equivalent
Communicated by V. A. Zinoviev. This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), Academic fund for outstanding talents in universities (gxbjZD03).
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Minjia Shi [email protected] Li Xu [email protected] Patrick Solé [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
CNRS, University of Aix Marseille, Centrale Marseille, I2M, Marseille, France
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to their duals [1,7,17,21]. These constitute in turn a subclass of the larger class of formally self-dual codes that is to say codes the weight enumerator of which is a fixed point of the MacWilliams transform [6,8–10]. A very popular and successful construction technique for isodual codes is the use of circulant matrices. In particular double circulant codes are easily shown to be isodual [5]. In the present paper, we generalize this technique to polycirculant matrices, and introduce double polycirculant codes. In [2,20] was studied the notion of polycyclic codes, that is linear codes over a finite field F, that are invariant under a generalized shift (called here a polyshift), and affording a structure of ideal over a ring of the form R f = F[x]/ f for some f ∈ F[x] (the case f = x n − 1 is that of classical cyclic codes). While the name was coined in [20], the concept (under the name pseudo-cyclic code) has been known for a long time [23]. As is well known, polycyclic codes are shortened cyclic codes, and conversely shortened cyclic codes are polycyclic [23, p .241]. In the present paper,
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