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Enumeration formulae for self-orthogonal, self-dual and complementary-dual additive cyclic codes over finite commutative chain rings

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Abstract

Let R, S be two finite commutative chain rings such that R is the Galois extension of S of degree \(r \ge 2\) and has a self-dual basis over S. Let q be the order of the residue field of S,  and let N be a positive integer with \(\gcd (N,q)=1.\) An S-additive cyclic code of length N over R is defined as an S-submodule of \(R^N,\) which is invariant under the cyclic shift operator on \(R^N.\) In this paper, we show that each S-additive cyclic code of length N over R can be uniquely expressed as a direct sum of linear codes of length r over certain Galois extensions of the chain ring S,  which are called its constituents. We further study the dual code of each S-additive cyclic code of length N over R by placing the ordinary trace bilinear form on \(R^N\) and relating the constituents of the code with that of its dual code. With the help of these canonical form decompositions of S-additive cyclic codes of length N over R and their dual codes, we further characterize all self-orthogonal, self-dual and complementary-dual S-additive cyclic codes of length N over R in terms of their constituents. We also derive necessary and sufficient conditions for the existence of a self-dual S-additive cyclic code of length N over R and count all self-dual and self-orthogonal S-additive cyclic codes of length N over R by considering the following two cases: (I) both q, r are odd, and (II) q is even and \(S=\mathbb {F}_{q}[u]/\langle u^e \rangle .\) Besides this, we obtain the explicit enumeration formula for all complementary-dual S-additive cyclic codes of length N over R. We also illustrate our main results with some examples.

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References

  1. Bágio, D., Dias, I., Paques, A.: On self-dual normal bases. Indag. Math. 17(1), 1–11 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Betty, R.A., Munemasa, A.: A mass formula for self-orthogonal codes over \(\mathbb{Z} _{p^2}\). J. Combinator. Inform. Syst. Sci. 34, 51–66 (2009)

    MATH  Google Scholar 

  3. Bini, G., Flamini, F.: Finite commutative rings and their applications. Kluwer Academic Publishers, Boston (2002)

    Book  MATH  Google Scholar 

  4. Calderbank, A.R., Rains, E.M., Shor, P.M., Sloane, N.J.: Quantum error correction via codes over \(GF(4)\). IEEE Trans. Inform. Theory 44(4), 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calderbank, A.R., Sloane, N.J.: Modular and \(p\)-adic cyclic codes. Des. Codes Cryptogr. 6, 21–35 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, Y., Gao, J., Fu, F.W., Cao, Y.: Enumeration and construction of additive cyclic codes over Galois rings. Discrete Math. 338(6), 922–937 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dougherty, S.T., Kim, J.L., Liu, H.: Constructions of self-dual codes over finite commutative chain rings. Int. J. Inf. Coding Theory 1(2), 171–190 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Grove, L.C.: Classical groups and geometric algebra. American Mathematical Society, Providence, Rhode Island (2008)

    MATH  Google Scholar 

  9. Guenda, K. and Gulliver, T.A.: Quantum codes over rings. Int. J. Quantum Inf. 12(04), Article no. 1450020 (2014)

  10. Hammons Jr, A.R., Kumar, P.V., Calderbank, A. R., Sloane, N.J.A., Solé, P.: The \(\mathbb{Z}_4\)-linearity Kerdock. Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40, 301-319 (1994)

  11. Honold, T., Landjev, I.: Linear codes over finite chain rings. Electron. J. Combin. 7, Article no. R11, (2000)

  12. Honold, T., Landjev, I.: Linear codes over finite chain rings and projective Hjelmslev geometries. Proc. CIMPA Summer School - Codes over rings 6, 60–123 (2009)

  13. Huffman, W.C.: Additive cyclic codes over \(\mathbb{F} _4\). Adv. Math. Commun. 1(4), 427–459 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huffman, W.C.: Additive cyclic codes over \(\mathbb{F} _4\). Adv. Math. Commun. 2(3), 309–343 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huffman, W.C.: Cyclic \(\mathbb{F} _q\)-linear \(\mathbb{F} _{q^t}\)-codes. Int. J. Inf. Coding Theory 1(3), 249–284 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Huffman, W.C.: On the theory of \(\mathbb{F} _q\)-linear \(\mathbb{F} _{q^t}\)-codes. Adv. Math. Commun. 7(3), 349–378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jitman, S., Udomkavanich, P.: The Gray image of codes over finite chain rings. Int. J. Contemp. Math. Sciences 5(10), 449–458 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Kaur, T., Sharma, A.: Constacyclic additive codes over finite fields. Discrete Math. Algorithms Appl. 9(3), Article no. 1750037, (2017)

  19. Lang, S.: Undergraduate algebra. Springer-Verlag, New York (1990)

    Book  MATH  Google Scholar 

  20. Mahmoudi, S., Samei, K.: Cyclic \(R\)-additive codes. Discrete Math. 340, 1657–1668 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mahmoudi, S., Samei, K.: Additive codes over Galois rings. Finite Fields Appl. 56, 332–350 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. McDonald, B.R.: Finite rings with identity. Marcel Dekker, New York (1974)

    MATH  Google Scholar 

  23. Martinez-Moro, E., Otal, K., Özbudak, F.: Additive cyclic codes over finite commutative chain rings. Discrete Math. 341(7), 1873–1884 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Norton, G.H., Sălăgean, A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra Engrg. Comm. Comput. 10, 489–506 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peterson, W.W., Brown, D.T.: Cyclic codes for error detection. Proceedings of the IRE 49(1), 228–235 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pless, V.: On the uniqueness of the Golay codes. J. Comb. Theory 5(3), 215–228 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  27. Seroussi, G., Lempel, A.: Factorization of symmetric matrices and trace-orthogonal bases in finite fields. SIAM J. Comput. 9(4), 758–767 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sharma, A., Kaur, T.: On cyclic \(\mathbb{F} _q\)-linear \(\mathbb{F} _{q^t}\)-codes. Int. J. Inf. Coding Theory 4(1), 19–46 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Sharma, A., Kaur, T.: Enumeration formulae for self-dual, self-orthogonal and complementary-dual quasi-cyclic codes over finite fields. Cryptogr. Commun. 10, 401–35 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sharma, A., Kaur, T.: Enumeration of complementary-dual cyclic \(\mathbb{F} _q\)-linear \(\mathbb{F} _{q^t}\)-codes. Discrete Math. 341(4), 965–980 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sidana, T., Kashyap, N.: Entanglement-assisted quantum error-correcting codes over local Frobenius rings, arXiv:2202.00248 (2022)

  32. Taylor, D.E.: The geometry of the classical groups, Sigma Series in Pure Mathematics, Heldermann Verlag (1992)

  33. Wan, Z.X.: Lectures on finite fields and Galois rings. World Scientific Publishing, New Jersey (2012)

    MATH  Google Scholar 

  34. Yadav, M., Sharma, A.: Mass formulae for Euclidean self-orthogonal and self-dual codes over finite commutative chain rings. Discrete Math. 344(1), 1–24 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yadav, M., Sharma, A.: On the enumeration and classification of \(\sigma \)-LCD codes over finite commutative chain rings. Discrete Math. 345(8), 1–25 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yadav, M., Sharma, A.: A recursive method for the construction and enumeration of self-orthogonal and self-dual codes over the quasi-Galois ring \(\mathbb{F} _{2^r}[u]/\langle u^e\rangle \). Des. Codes Cryptogr. 91(5), 1973–2003 (2023)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Leijo Jose research support by the National Board for Higher Mathematics (NBHM), India, under Grant no. 0203/13(46)/2021-R &DII/13176, is gratefully acknowledged. Anuradha Sharma research support by the Department of Science and Technology, India, under the Grant no. DST/INT/RUS/RSF/P-41/2021 with TPN 65025, is gratefully acknowledged.

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  1. Department of Mathematics, IIIT-Delhi, New Delhi, 110020, India

    Leijo Jose & Anuradha Sharma

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  1. Leijo Jose
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Jose, L., Sharma, A. Enumeration formulae for self-orthogonal, self-dual and complementary-dual additive cyclic codes over finite commutative chain rings. Cryptogr. Commun. 16, 1383–1416 (2024). https://doi.org/10.1007/s12095-024-00728-9

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