On codes over product of finite chain rings

Abstract

In this paper, codes over the direct product of two finite commutative chain rings are studied. The standard form of the parity-check matrix is determined. The structure of self-dual codes is described. A distance preserving Gray map from the direct product of chain rings to a finite field is defined. Two upper bounds on minimum distance are obtained.

Keywords

• MDS Codes
• Self-Dual Codes
• Chain Rings

References

1. REFERENCES
2. [1] Aydogdu I. Codes over Zp[u]/〈ur〉 × Zp[u]/〈us〉. J Algebra Comb Discrete Struct Appl 2019;32:39–51. [CrossRef]
3. [2] Aydogdu I, Abualrub T, Siap, I. On 22-additive codes. Int J Comput Math 2014;92:1806–1814. [CrossRef]
4. [3] Aydogdu I, Siap I. On pprs-additive codes. Linear Multilinear Algebra 2014;63:2089–2102. [CrossRef]
5. [4] Aydogdu I, Siap I, Ten-Valls R. On the structure of 22u3 -linear and cyclic codes. Finite Fields Their Appl 2017;48:241–260. [CrossRef]
6. [5] Bilal M, Borges J, Dougherty ST, Fernández-Córdoba C. Maximum distance separable codes over 42 ´ and ´42. Des Codes Cryptogr 2010;61:31=40.
7. [6] Borges J, Fernández-Córdoba C. A characterization of 22[]u -linear codes. Des Codes Cryptogr 2017; 86:1377–1389. [CrossRef]
8. [7] Borges J, Fernandez-Cordoba C, Ten-Valls R. 24- additive cyclic codes, generator polynomials, and dual codes. IEEE Trans Inf Theory 2016;62:6348–6354. [CrossRef]

9. [8] Borges J, Fernández-Córdoba C, Ten-Valls R. Linear and cyclic codes over direct product of finite chain rings. Math Methods Appl Sci 2017;41:6519–652. [CrossRef]
10. [9] Constantinescu I, Heise W. A metric for codes over residue class rings of integers. Probl Inf Transm 1997;33:22–28.
11. [10] Dinh HQ, Bag T, Kewat PK, Pathak S, Upadhyay AK, Chinnakum W. Constacyclic codes of length (pr, ps) over mixed alphabets. J Appl Math Comput 2021;67:807–832. [CrossRef]
12. [11] Diao L, Gao J, Lu J. Some results on ppv[]-addi-tive cyclic codes. Adv Math Commun 2020;14:555–572. [CrossRef]
13. [12] Dinh HQ, Pathak S, Bag T, Upadhyay AK, Chinnakum W. A study of qR-cyclic codes and their applications in constructing quantum codes. IEEE Access 2020;8:190049–190063. [CrossRef]
14. [13] Dinh HQ, Pathak S, Upadhyay AK, Yamaka W. New DNA codes from cyclic codes over mixed alphabets. Mathematics 2020;8:1977. [CrossRef]
15. [14] Dougherty ST. Algebraic Coding Theory Over Finite Commutative Rings. 1st ed. Berlin: Springer; 2017.[CrossRef]
16. [15] Dougherty ST, Kim JL, Kulosman H. MDS codes over finite principal ideal rings. Des Codes Cryptogr 2008;50:77–92. [CrossRef]
17. [16] Dougherty ST, Kim JL, Kulosman H, Liu H. Self-dual codes over commutative Frobenius rings. Finite Fields Their Appl 2010;16:14–26. [CrossRef]
18. [17] Dougherty ST, Kim JL, Liu H. Constructions of self-dual codes over finite commutative chain rings. Int J Inf Cod Theory 2010;1:171. [CrossRef]
19. [18] Gao J, Diao L. pp[]u -additive cyclic codes. Int J Inf Cod Theory 2018;5:1–17. [CrossRef]
20. [19] Grassl M. Bounds on the minimum distance of lin-ear codes and quantum codes. Available at: http://www.codetables.de. Accessed on May 22, 2023. [CrossRef]
21. [20] Greferath M, Schmidt SE. Gray isometries for finite chain rings and a nonlinear ternary code. IEEE Trans Inf Theory 1999;45:2522–2524. [CrossRef]
22. [21] Jitman J, Udomkavanich P. The gray image of codes over finite chain rings. Int J Contemp Math Sci 2010;5:449–458.
23. [22] Li J, Gao J, Fu FW, Ma F. qR-linear skew consta-cyclic codes and their application of constructing quantum codes. Quantum Inf Process 2020;19:193. [CrossRef]
24. [23] MacWilliams F, Sloane N. The Theory of Error-Correcting Codes. 1st ed. Amsterdam: North-Holland; 1977.
25. [24] Mahmoudi S, Samei K. SR-additive codes. Bull Korean Math Soc 2019;56:1235–1255. [CrossRef] [25] Melakhessou A, Aydin N, Hebbache Z, Guenda K. q+()uq -linear skew constacyclic codes. J Algebra Comb Discrete Struct Appl 2020;7:85–101. [CrossRef]
26. [26] Norton GH, Sălăgean A. On the structure of lin-ear and cyclic codes over a finite chain ring. Appl Algebra Eng Commun Comput 2000;10:489–506. [CrossRef]
27. [27] Rifa J, Pujol J. Translation-invariant propelinear codes. IEEE Trans Inf Theory 1997;43:590–598. [CrossRef]
28. [28]Samei K, Mahmoudi S. Singleton bounds for R- additive codes. Adv Math Commun 2018;12:107-114.
29. [29] Shi M, Wu R, Krotov DS. On ppk -additive codes and their duality. IEEE Trans Inf Theory 2019;65:3841–3847. [CrossRef]
30. [30] Wu R, Shi M. Some classes of mixed alphabet codes with few weights. IEEE Commun Lett 2021;25:1431–1434. [CrossRef]