MELAKHESSOU, Ahlem (2020) Codes sur les anneaux. Doctoral thesis, Université de Batna 2.
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Abstract
This work has reached this level by producing two journal papers and four conference papers. The first journal paper is entitled, « On Codes over Fq+vFq+v2Fq », it appeared in Journal of Applied Mathematics and Computing, 2017. In this paper we investigated linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = Fq+vFq+v2Fq, where v3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over Fq and extend these to codes over R. The second journal paper is entitled, « Zq(Zq + uZq)−Linear Skew Constacyclic Codes », it appeared in Journal of Algebra Combinatorics Discrete Structures and Applications. In this paper we study skew constacyclic codes over the ring ZqR, where R = Zq + uZq, q = ps for a prime p and u2 = 0. We give the definition of these codes as subsets of the ring Z�q R�. Some structural properties of the skew polynomial ring R[x, �] are discussed, where � is an automorphism of R. We describe the generator polynomials of skew constacyclic codes over ZqR, also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over ZqR we obtained some new linear codes over Z4. Finally, we have generalized these codes to double skew constacyclic codes over ZqR. The third paper is entitled, « Formally Self-dual Codes over Ak », it was presented at CMA- 2014 (Tlemcen). In this paper we present several kinds of construction of formally self-dual codes over the ring Ak = F2 [v1, . . . , vk] / hv2 i = vi, vivj = vjvii. The fourth paper is entitled, « LCD Codes over Fq +vFq +v2Fq », it was presented at CMA- 2016 (Batna). The purpose of this work is to investigate linear codes with complementary dual(LCD) codes over the ring R = Fq + vFq + v2Fq, where v3 = v, for q odd. The fifth paper is entitled, « Zq(Zq + vZq + . . . + vm−1Zq)− Linear Cyclic, Skew Cyclic and Constacyclic Codes », it was presented at ECMI-SciTech’2017 (Constantine). In this paper, we study cyclic, skew cyclic and constacyclic codes over the ring Zq(Zq + vZq + . . . + vm−1Zq), where q = ps, p is a prime and vm = v. We give the definition of these codes as subsets of the ring Z�q R�. The sixth paper is entitled, « Double Skew (1+u)−Constacyclic Codes over Z4(Z4+uZ4) », it was presneted at IWCA-2019 (Oran). In this paper, we study skew constacyclic codes over the ring Z4R where R = Z4 + uZ4, for u2 = 0. We give the definition of these codes as subsets of the ring Z�4 R�. Further, we have generalized these codes to double skew (1+u)−constacyclic codes over Z4R. Keywords: Formally self-dual codes, LCD codes, optimal codes, Gray map, automorphism, skew constacyclic codes, skew polynomial rings, double skew constacyclic codes.
| Item Type: | Thesis (Doctoral) |
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| Subjects: | Mathématiques |
| Divisions: | Faculté des mathématiques et de l'informatique > Département des mathématiques |
| Date Deposited: | 06 Jan 2021 10:34 |
| Last Modified: | 06 Jan 2021 10:34 |
| URI: | http://eprints.univ-batna2.dz/id/eprint/1891 |
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