Abstract
In this work we consider repeated-root multivariable codes over a finite chain ring. We show conditions for these codes to be principally generated. We consider a suitable set of generators of the code and compute its minimum distance. As an application we study the relevant example of the generalized Kerdock code in its r-dimensional cyclic version.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bini G and Flamini F (2002). Finite commutative rings and their applications. The Kluwer International Series in Engineering and Computer Science, 680. Kluwer Academic Publishers, Boston, MA
Cazaran J and Kelarev AV (1997). Generators and weights of polynomial codes. Arch Math 69: 479–486
Cazaran J and Kelarev AV (1999). On finite principal ideal rings. Acta Math Univ Comenianae LXVIII: 77–84
Cox D, Little J and O’Shea D (1992). Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer-Verlag, New York, Inc.
Dinh HQ and López-Permouth SR (2004). Cyclic and negacyclic codes over finite chain rings. IEEE Trans Inform Theory 50(8): 1728–1744
Hammons AR Jr, Kumar PV, Calderbank AR, Sloane NJA and Sole P (1994). The Z 4-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40: 301–319
Kerdock AM (1972). A class of low-rate non-linear binary codes. Inform Control 20: 182–187
Kurakin VL, Kuzmin AS, Markov VT, Mikhalev AV, Nechaev AA (1999) Linear codes and polylinear recurrences over finite rings and modules (a survey). In Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), vol 1719 of Lecture Notes in Comput Sci, pp 365–391. Springer, Berlin
Kuzmin AS, Nechaev AA (1994) Linearly presented codes and Kerdock code over an arbitrary Galois field of the characteristic 2. Russian Math Surveys 49(5)
Lu P, Liu M and Oberst U (2004). Linear recurring arrays, linear systems and multidimensional cyclic codes over quasi-Frobenius rings. Acta Appl Math 80(2): 175–198
Martínez-Moro E and Rúa IF (2006). Multivariable codes over finite chain rings: serial codes. SIAM on Discrete Mathematics 20(4): 947–959
McDonald BR (1974). Finite rings with identity. Marcel Dekker Inc, New York
Nechaev AA (1989). Kerdock’s code in a cyclic form. Diskret Mat 1: 123–139
Nechaev AA, Kuzmin AS (1997) Formal duality of linearly presentable codes over a Galois field. In Applied algebra, algebraic algorithms and error-correcting codes (Toulouse, 1997), vol 1255 of Lecture Notes in Comput Sci, pp 263–276. Springer, Berlin
Nechaev AA and Mikhailov DA (2001). Canonical system of generators of a unitary polynomial ideal over a commutative Artinian chain ring. Discrete Math Appl 11(6): 545–586
Puninski G (2001). Serial rings. Kluwer Academic Publishers, Dordrecht
Sălăgean A (2006). Repeated-root cyclic and negacyclic codes over a finite chain ring. Discrete Appl Math 154: 413–419
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: D. Jungnickel.
Rights and permissions
About this article
Cite this article
Martínez-Moro, E., Rúa, I.F. On repeated-root multivariable codes over a finite chain ring. Des. Codes Cryptogr. 45, 219–227 (2007). https://doi.org/10.1007/s10623-007-9114-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-007-9114-1