On a class of repeated-root monomial-like abelian codes. (English) Zbl 1371.94686
Summary: In this paper we study polycyclic codes of length \(p^{s_1} \times \cdots \times p^{s_n}\) over \(\mathbb F_{p^a}\) generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of J. Massey et. al. [IEEE Trans. Inf. Theory 19, No. 1, 101–110 (1973; Zbl 0248.94009)] on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.
MSC:
| 94B15 | Cyclic codes |
| 94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
Citations:
Zbl 0248.94009References:
| [1] | Castagnoli, G.; Massey, J. L.; Schoeller, P. A.; von Seemann, N.;, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37, 2, 337-342 (1991) · Zbl 0721.94016 |
| [2] | Dinh, H. Q.;, On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Fields Appl., 14, 1, 22-40 (2008) · Zbl 1129.94047 |
| [3] | Drensky, V.; Lakatos, P.;, Monomial ideals, group algebras and error correcting codes, In Applied algebra, algebraic algorithms and error-correcting codes, (Rome, 1988), Lecture Notes in Comput. Sci., Springer, Berlin, 357, 181-188 (1989) · Zbl 0679.16006 |
| [4] | Goldschmidt, D. M.;, Algebraic functions and projective curves, Graduate Texts in Mathematics, Springer-Verlag, New York, 215 (2003) · Zbl 1034.14011 |
| [5] | Hirschfeld, J. W. P.; Korchmáros, G.; Torres, F.;, Algebraic curves over a finite field, Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2008) · Zbl 1200.11042 |
| [6] | Huffman, W. C.; Pless, V.;, Fundamentals of error-correcting codes, Cambridge University Press, Cambridge (2003) · Zbl 1099.94030 |
| [7] | López-Permouth, S. R.; Szabo, S.;, On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings, Adv. Math. Commun., 3, 4, 409-420 (2009) · Zbl 1196.94085 |
| [8] | Martínez-Moro, E.; Rúa, I. F.;, On repeated-root multivariable codes over a finite chain ring, Des. Codes Cryptogr., 45, 2, 219-227 (2007) · Zbl 1196.94088 |
| [9] | Martínez-Pérez, C.; Willems, W.;, On the weight hierarchy of product codes, Des. Codes Cryptogr., 33, 2, 95-108 (2004) · Zbl 1077.94023 |
| [10] | Massey, J. L.; Costello, D. J.; Justesen, J.;, Polynomial weights and code constructions, IEEE Trans. Information Theory, IT-19, 101-110 (1973) · Zbl 0248.94009 |
| [11] | Özadam, H.; Özbudak, F.;, A note on negacyclic and cyclic codes of length \(p^s\) over \(a\) finite field of characteristic \(p\), Adv. Math. Commun., 3, 3, 265-271 (2009) · Zbl 1232.94019 |
| [12] | López-Permouth, S. R.; Özadam, H.; Özbudak, F.; Szabo, S.;, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields and Their Applications, 19, 1, 16-38 (2013) · Zbl 1272.94092 |
| [13] | Schaathun, H. G.;, The weight hierarchy of product codes, IEEE Trans. Inform. Theory, 46, 7, 2648-2651 (2000) · Zbl 1001.94053 |
| [14] | van Lint, J. H.;, Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37, 2, 343-345 (1991) · Zbl 0721.94017 |
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