Abstract
We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring \(R=\mathbb {F}_q[x]/\langle x^n+1\rangle , \) when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of \(x^n+1\) over \(\mathbb {F}_q, \) with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q, we show these conditions are met for infinitely many n’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov–Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q.
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Communicated by D. Panario.
This research is supported by National Natural Science Foundation of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11), Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008, gxyqZD2016270) and the Project of Graduate Academic Innovation of Anhui University (No. yfc 100015).
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Shi, M., Qian, L. & Solé, P. On self-dual negacirculant codes of index two and four. Des. Codes Cryptogr. 86, 2485–2494 (2018). https://doi.org/10.1007/s10623-017-0455-0
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DOI: https://doi.org/10.1007/s10623-017-0455-0