On a class of repeated-root monomial-like abelian codes

Öz

In this paper we study polycyclic codes of length $p^{s_1} \times \cdots \times p^{s_n}$\ over $\F_{p^a}$\ 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 Massey et. al. in \cite{MASSEY_1973} on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.

Anahtar Kelimeler

• Repeated-root Cyclic code
• Abelian code
• Weight-retaining property

Kaynakça

1. G. Castagnoli, J. L. Massey, P. A. Schoeller, and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37(2), 337-342, 1991.
2. H. Q. Dinh. On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Fields Appl., 14(1), 22-40, 2008.
3. V. Drensky and P. Lakatos, Monomial ideals, group algebras and error correcting codes, In Applied algebra, algebraic algorithms and error-correcting codes, (Rome, 1988), volume 357 of Lecture Notes in Comput. Sci., pages 181-188. Springer, Berlin, 1989.
4. D. M. Goldschmidt, Algebraic functions and projective curves, volume 215, Graduate Texts in Math- ematics, Springer-Verlag, New York, 2003.
5. J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, Algebraic curves over a finite field, Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2008. W. C. Huffman, V. Pless,
6. Fundamentals of error-correcting codes, Cambridge University Press, Cambridge, 2003.
7. S. R. López-Permouth, S. Szabo, On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings, Adv. Math. Commun., 3(4), 409-420, 2009.
8. E. Martínez-Moro, I. F. Rşa, On repeated-root multivariable codes over a finite chain ring, Des. Codes Cryptogr., 45(2), 219-227, 2007.

9. C. Martínez-Pérez, W. Willems, On the weight hierarchy of product codes, Designs, Codes and Cryptography. An International Journal, 33(2), 95-108, 2004.
10. J. L. Massey, D. J. Costello, Jİrn Justesen, Polynomial weights and code constructions, IEEE Trans. Information Theory, IT-19, 101-110, 1973.
11. Hakan Özadam and Ferruh Özbudak, A note on negacyclic and cyclic codes of length psover a finite field of characteristic p, Adv. Math. Commun., 3(3), 265-271, 2009.
12. S. R. López-Permouth, H. Özadam, F. Özbudak, S Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields and Their Applications, 19(1), 16-38, 20 H. G. Schaathun, The weight hierarchy of product codes, IEEE Trans. Inform. Theory, 46(7), 2648- 2651, 2000.
13. J. H. van Lint. Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37(2), 343-345, 1991.