Abstract
We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.
This is a preview of subscription content, log in via an institution to check access.
References
Yu. L. Vasil’ev, “On Nongroup Close-Packed Codes,” in Problems of Cybernetics, Vol. 8, Ed. by A. A. Lyapunov (Fizmatgiz, Moscow, 1962), pp. 375–378.
A. M. Romanov, “On Construction of Nonlinear Perfect Binary Codes by Inversion of Symbols,” Diskretn. Anal. Issled. Oper. Ser. 1, 4 (1), 46–52 (1997).
A. M. Romanov, “On Partitions of q-Ary Hamming Codes into Disjoint Components,” Diskretn. Anal. Issled. Oper. Ser. 1, 11 (3), 80–87 (2004).
A. M. Romanov, “A Survey of Methods for Constructing Nonlinear Perfect Binary Codes,” Diskretn. Anal. Issled. Oper. Ser. 1, 13 (4), 60–88 (2006) [J. Appl. Indust. Math. 2 (2), 252–269 (2008)].
A. M. Romanov, “On Admissible Families of Components of Hamming Codes,” Diskretn. Anal. Issled. Oper. 19, (2), 84–91 (2012) [J. Appl. Indust. Math. 6 (3), 355–359 (2012)].
S. V. Avgustinovich and D. S. Krotov, “Embedding in a Perfect Code,” J. Combin. Des. 17 (5), 419–423 (2009).
T. Etzion and A. Vardy, “Perfect Binary Codes: Constructions, Properties, and Enumeration,” IEEE Trans. Inform. Theory. 40 (3), 754–763 (1994).
T. Etzion, “Nonequivalent q-Ary Perfect Codes,” SIAMJ. DiscreteMath. 9 (3), 413–423 (1996).
O. Heden and D. S. Krotov, “On the Structure of Non-Full-Rank Perfect q-Ary Codes,” Adv. Math. Combin. 5 (2), 149–156 (2011).
B. Lindström, “On Group and Nongroup Perfect Codes in q Symbols,” Math. Scand. 25, 149–158 (1969).
A. V. Los’, “Construction of Perfect q-Ary Codes,” in Proceedings of 9th I, Kranevo, Bulgaria, June 19–25, 2004 (Inst. Math. Inform., Sofia, 2004), pp. 272–276.
P. R. J. Östergrd, O. Pottonen, and K. T. Phelps, “The Perfect Binary One-Error-Correcting Codes of Length 15: Part II—Properties,” IEEE Trans. Inform. Theory 56 (6), 2571–2582 (2010).
K. T. Phelps and M. Villanueva, “Ranks of q-Ary 1-Perfect Codes,” Design. Codes Cryptogr. 27 (1–2), 139–144 (2002).
K. T. Phelps, J. Rifà, and M. Villanueva, “Kernels and p-Kernels of p r-Ary 1-Perfect Codes,” Design. Codes Cryptogr. 37 (2), 243–261 (2005).
A. M. Romanov, “Hamiltonicity of Minimum Distance Graphs of 1-Perfect Codes,” Electron. J. Combin. 19 (1), P65, 1–6 (2012).
J. Schönheim, “On Linear and Nonlinear Single-Error-Correcting q-Ary Perfect Codes,” Inform. Control 12, 23–26 (1968).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.M. Romanov, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 3, pp. 107–123.
Rights and permissions
About this article
Cite this article
Romanov, A.M. On full-rank perfect codes over finite fields. J. Appl. Ind. Math. 10, 444–452 (2016). https://doi.org/10.1134/S1990478916030157
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478916030157