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On equivalence of cyclic codes, generalization of a quasi-twisted search algorithm, and new linear codes

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Abstract

A fundamental problem in coding theory is the explicit construction of linear codes with best possible parameters. A search algorithm (ASR) on certain types of quasi-twisted (QT) codes has been very fruitful to address this challenging problem. In this work, we generalize the ASR algorithm to make it more comprehensive. The generalization is based on code equivalence. As a result of implementing the more general algorithm, we discovered 27 new linear codes over the fields \(\mathbb {F}_q\) for \(q=3,4,5,\) and 7. Further, we prove several useful theoretical results about the equivalence of cyclic codes, constacyclic codes, and QT codes.

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Notes

  1. Note that when we use the standard generator for a constacyclic code, the shift constant does not show up in the first coordinate of each row vector because they are all zeros. If we use a generator polynomial of a higher degree however, then it will be necessary to include the shift constant a.

References

  1. Ackerman R., Aydin N.: New quinary linear codes from quasi-twisted codes and their duals. Appl. Math. Lett. 24(4), 512–515 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  2. Aydin N., Murphree J.: New linear codes from constacyclic codes. J. Frankl. Inst. 351(3), 1691–1699 (2014).

    Article  MATH  Google Scholar 

  3. Aydin N., Siap I.: New quasi-cyclic codes over \({\mathbb{F}}_5\). Appl. Math. Lett. 15(7), 833–836 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  4. Aydin N., Siap I., Ray-Chaudhuri D.K.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24(3), 313326 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. Aydin N,. Asamov T., Gulliver, T.A.: Some open problems on quasi-twisted and related code constructions and good quaternary codes. In: Proceeding of International Symposium on Information Theory Nice, France, pp. 856–860 (2007).

  6. Aydin N., Connolly N., Murphree J.: New binary linear codes from quasi-cyclic codes and an augmentation algorithm. Appl. Algebra Eng. Commun. Comput. 28(4), 339–350 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  7. Aydin N., Connolly N., Grassl M.: Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes. Adv. Math. Commun. 11(1), 245–258 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  8. Babai L., Codenotti P., Groshow, J.A., Qiao, Y.: Code equivalence and group isomorphism. In: Proceeding of ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, 13951408 (2011).

  9. Berlekamp E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968).

    MATH  Google Scholar 

  10. Chen Z.: Six new binary quasi-cyclic codes. IEEE Trans. Inf. Theory 40(5), 1666–1667 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  11. Daskalov R., Gulliver T.A.: New quasi-twisted quaternary linear codes. IEEE Trans. Inf. Theory 46(7), 2642–2643 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  12. Daskalov R., Hristov P.: New binary one-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 49(11), 3001–3005 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  13. Daskalov R., Hristov P.: New quasi-twisted degenerate ternary linear codes. IEEE Trans. Inf. Theory 49(9), 2259–2263 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  14. Daskalov R., Hristov P., Metodieva E.: New minimum distance bounds for linear codes over \(GF(5)\). Discret. Math. 275(1), 97–110 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  15. Dobson E.: Monomial isomorphisms of cyclic codes. Des. Codes Cryptogr. 76(2), 257–267 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  16. Grassl M.: Code tables: bounds on the parameters of codes. http://www.codetables.de/.

  17. Guenda K., Gulliver T.A.: On the equivalence of cyclic and quasi-cyclic codes over finite fields. J. Algebra Comb. Discret. Struct. Appl. 4(3), 261–269 (2017).

    MathSciNet  Google Scholar 

  18. Gulliver T.A., Bhargava V.K.: New good rate \((m1)/pm\) ternary and quaternary quasi-cyclic codes. Des. Codes Cryptogr. 7(3), 223–233 (1996).

    MathSciNet  MATH  Google Scholar 

  19. Huffman W.C.: The equivalence of two cyclic objects on \(pq\) elements. Discret. Math. 154(1), 103–127 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  20. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  21. Huffman W.C., Job V., Pless V.: Multipliers and generalized multipliers of cyclic objects and cyclic codes. J. Comb. Theory A 62(2), 183–215 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  22. Leon J.S., Masley J.M., Pless V.: Duadic codes. IEEE Trans. Inf. Theory 30(5), 709–714 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  23. Magma computer algebra system. http://magma.maths.usyd.edu.au/.

  24. McEliece R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Prog. Rep. 42–44, 114–116 (1978).

    Google Scholar 

  25. Muzychuk M.: A solution of an equivalence problem for semisimple cyclic codes. arXiv:1105.4320v1 (2011).

  26. Newhart D.W.: Inequivalent cyclic codes of prime length. IEEE Trans. Inf. Theory 32(5), 703–705 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  27. Petrank E., Roth R.M.: Is code equivalence easy to decide? IEEE Trans. Inf. Theory 43(5), 1602–1604 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  28. Prange E.: Cyclic Error-Correcting Codes in Two Symbols. Technical Report TN-57-103. Air Force Cambridge Research Center, Cambridge (1957).

    Google Scholar 

  29. Prange E.: Some Cyclic Error-Correcting Codes with Simple Decoding Algorithm. Technical Report TN-58-156. Air Force Cambridge Research Center, Cambridge (1958).

    Google Scholar 

  30. Sendrier N.: Finding the permutation between equivalent linear codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46(4), 1193–1203 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  31. Sendrier N., Simos D.E.: How easy is code equivalence over \(\mathbb{F}_q\)? In: Proceeding of International Workshop on Coding Theory and Crypt, Bergen, Norway (2013).

  32. Sendrier N., Simos D.E.: The hardness of code equivalence over \(\mathbb{F}_q\) and its application to code-based cryptography. In: Gaborit P. (ed.) Post-quantum Cryptography, vol. 7932, pp. 203–216. Springer Lecture Notes in Computer ScienceSpringer, Limoges (2013).

    Chapter  Google Scholar 

  33. Siap I., Aydin N., Ray-Chaudhuri D.K.: New ternary quasi-cyclic codes with improved minimum distances. IEEE Trans. Inf. Theory 46(4), 1554–1558 (2000).

    Article  MATH  Google Scholar 

  34. Vardy A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory 43(6), 1757–1766 (1997).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Kenyon College, Gambier, OH, 43022, USA

    Nuh Aydin, Jonathan Lambrinos & Oliver VandenBerg

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  1. Nuh Aydin
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  2. Jonathan Lambrinos
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  3. Oliver VandenBerg
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Correspondence to Nuh Aydin.

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Communicated by V. A. Zinoviev.

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Aydin, N., Lambrinos, J. & VandenBerg, O. On equivalence of cyclic codes, generalization of a quasi-twisted search algorithm, and new linear codes. Des. Codes Cryptogr. 87, 2199–2212 (2019). https://doi.org/10.1007/s10623-019-00613-0

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