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Linear Block and Convolutional MDS Codes to Required Rate, Distance and Type

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Intelligent Computing (SAI 2022)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 507))

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Abstract

Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required error-correcting capability and required types. Infinite series of block codes with rate approaching a given rational R with \(0<R<1\) and relative distance over length approaching \((1-R)\) are designed. These can be designed over fields of given characteristic p or over fields of prime order and can be specified to be of a particular type such as (i) dual-containing under Euclidean inner product, (ii) dual-containing under Hermitian inner product, (iii) quantum error-correcting, (iv) linear complementary dual (LCD). Convolutional codes to required rate and distance and infinite series of convolutional codes with rate approaching a given rational R and distance over length approaching \(2(1-R)\) are designed. The designs are algebraic and properties, including distances, are shown algebraically. Algebraic explicit efficient decoding methods are referenced.

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Notes

  1. 1.

    There exist very few algebraic constructions for designing convolutional codes and search methods limit their size and availability, see McEliece [4] for discussion and also [10,11,12,13].

  2. 2.

    This has different parameter requirements for linear block codes, convolutional codes and QECCs.

References

  1. Blahut, R.E.: Algebraic Codes for Data Transmission. Cambridge University Press (2003)

    Google Scholar 

  2. Johannesson, R., Zigangirov, K.: Fundamentals of Convolutional Coding. Wiley-IEEE Press (1999)

    Google Scholar 

  3. McEliece, R.J.: Theory of Information and Coding, 2nd edn. Cambridge University Press (2002)

    Google Scholar 

  4. McEliece, R.J.: The algebraic theory of convolutional codes. In: Handbook of Coding Theory, Volume I. Elsevier Science, North Holland (1998)

    Google Scholar 

  5. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier (1977)

    Google Scholar 

  6. Almheiri, A., Dong, X., Harlow, D.: Bulk locality and quantum error correction in AdS/CFT. arXiv arXiv:1411.7041 (2014)

  7. Bocharova, I., Hug, F., Johannesson, R., Kudryashov, B.: Dual convolutional codes and the MacWilliams identities. Probl. Inf. Transm. 48(1), 21–30 (2012)

    Article  MathSciNet  Google Scholar 

  8. Hurley, T.: Linear complementary dual, maximum distance separable codes. arXiv arXiv:1901.04241 (2020)

  9. Hurley, T., Hurley, D., Hurley, B.: Quantum error-correcting codes: the unit-derived strategy. Int. J. Inf. Coding Theor. 5(2), 169–182 (2018)

    MATH  Google Scholar 

  10. Almeida, P., Napp, D., Pinto, R.: A new class of superregular matrices and MDP convolutional codes. Linear Algebra Appl. 439(7), 2145–2157 (2013)

    Article  MathSciNet  Google Scholar 

  11. Almeida, P., Napp, D., Pinto, R.: Superregular matrices and applications to convolutional codes. Linear Algebra Appl. 499, 1–25 (2016)

    Article  MathSciNet  Google Scholar 

  12. Guardia, G.: On negacyclic MDS-convolutional codes. Linear Algebra Appl. 448(Suppl. C), 85–96 (2014)

    Article  MathSciNet  Google Scholar 

  13. Muñoz Porras, J., Domínguez Pérez, J., Iglesias, C.J., Serrano Sotelo, G.: Convolutional Goppa codes. IEEE Trans. Inf. Theor. 52(1), 340–344 (2006)

    Article  MathSciNet  Google Scholar 

  14. Carlet, C., Mesnager, S., Tang, C., Qi, Y: Euclidean and Hermitian LCD MDS codes. Des. Codes Crypt. 86(11), 2605–2618 (2018). arXiv:1702.08033 (2017)

  15. Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pelikaan, R.: Linear codes over \(F_q\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theor. 64(4), 3010–3017 (2018)

    Article  Google Scholar 

  16. Carlet, C., Mesnager, S., Tang, C., Qi, Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theor. 65, 39–49 (2018). arXiv:1709.03217 (2017)

  17. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010). Monograph Book

    Google Scholar 

  18. Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. In: Pinto, R., Rocha Malonek, P., Vettori, P. (eds.) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol. 3, pp. 97–105. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-17296-5_9. J. Adv. in Math. of Comm., 10(1), 131–150, 2016

  19. Massey, J.L.: Linear codes with complementary duals. Discrete Math. 105(106), 337–380 (1992)

    Article  MathSciNet  Google Scholar 

  20. Massey, J.L.: Reversible codes. Inf. Control 7(3), 369–380 (1964)

    Article  MathSciNet  Google Scholar 

  21. Mesnager, S., Tang, C., Qi, Y.: Complementary dual algebraic geometry codes. IEEE Trans. Inf. Theor. 64, 4 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Aly, S.A., Grassl, M., Klappenecker, A., Rötteler, M., Sarvepalli, P.K.: Quantum convolutional BCH codes. In: Proceedings of the IEEE 10th CWIT, pp. 180–183 (2007)

    Google Scholar 

  24. Gluesing-Luerssen, H., Helmke, U., Iglesias Curto, J.I.: Algebraic decoding for doubly cyclic convolutional codes. arXiv:0908.0753 (2009)

  25. Gluesing-Luerssen, H., Schneider, G.: A MacWilliams identity for convolutional codes: the general case. IEEE Trans. Inf. Theor. 55(7), 2920–2930 (2009)

    Article  MathSciNet  Google Scholar 

  26. Hurley, T.: Maximum distance separable codes to order. arXiv arXiv:1902.06624 (2019)

  27. Hurley, P., Hurley, T: Module codes in group rings In: ISIT 2007, Nice, pp. 1981–1985 (2007)

    Google Scholar 

  28. Hurley, B., Hurley, T.: Systems of MDS codes from units and idempotents. Discrete Math. 335, 81–91 (2014)

    Article  MathSciNet  Google Scholar 

  29. Hurley, T.: Convolutional codes from units in matrix and group rings. Int. J. Pure Appl. Math. 50(3), 431–463 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Rosenthal, J., Smarandache, R.: Maximum distance separable convolutional codes. Appl. Algebra Engrg. Comm. Comput. 10(1), 15–32 (1999)

    Article  MathSciNet  Google Scholar 

  31. Rosenthal, J.: Connections between linear systems and convolutional codes. In: Marcus, B., Rosenthal, J. (eds.) Codes, Systems, and Graphical Models, Minneapolis, New York, pp. 39–66 (1999)

    Google Scholar 

  32. Rosenthal, J.: An algebraic decoding algorithm for convolutional codes. In: Picci, G., Gilliam, D.S. (eds.) Dynamical Systems. Control, Coding, Computer Vision: New Trends, Interfaces, and Interplay, pp. 343–360. Birkhäuser, Boston-Basel-Berlin (1999)

    Google Scholar 

  33. Hurley, P., Hurley, T.: Codes from zero-divisors and units in group rings. Int. J. Inform. Coding Theor. 1, 57–87 (2009)

    Article  MathSciNet  Google Scholar 

  34. Hurley, P., Hurley, T: Block codes from matrix and group rings, chap. 5. In: Woungang, I., Misra, S., Misma, S.C. (eds.) Selected Topics in Information and Coding Theory, pp. 159–194. World Scientific (2010)

    Google Scholar 

  35. Hurley, P., Hurley, T.: LDPC and convolutional codes from matrix and group rings, chap. 6. In: Woungang, I., Misra, S., Misma, S.C. (eds.) Selected Topics in Information and Coding Theory, pp. 195–239. World Scientific (2010)

    Google Scholar 

  36. Hurley, T.: Convolutional codes from unit schemes. arXiv arXiv:1412.1695 (2020, revised)

  37. Hurley, T., Hurley, D.: Coding theory: the unit-derived methodology. Int. J. Inf. Coding Theor. 5(1), 55–80 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Rains, E.: Nonbinary quantum codes. IEEE Trans. Inf. Theor. 43, 1827–1832 (1999)

    Article  MathSciNet  Google Scholar 

  39. Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theor. 47(7), 3065–3072 (2001)

    Article  MathSciNet  Google Scholar 

  40. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theor. 52(11), 4892–4914 (2006)

    Article  MathSciNet  Google Scholar 

  41. Hurley, T., McEvoy, P., Wenus, J.: Algebraic constructions of LDPC codes with no short cycles. Int. J. Inf. Coding Theor. 1(3), 285–297 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Smarandache, R., Gluesing-Luerssen, H., Rosenthal, J.: Constructions for MDS-convolutional codes. IEEE Trans. Inf. Theor. 47, 2045–2049 (2001)

    Article  MathSciNet  Google Scholar 

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  1. National University of Ireland Galway, Galway, Ireland

    Ted Hurley

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  1. Ted Hurley
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Correspondence to Ted Hurley .

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  1. Saga University, Saga, Japan

    Kohei Arai

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Hurley, T. (2022). Linear Block and Convolutional MDS Codes to Required Rate, Distance and Type. In: Arai, K. (eds) Intelligent Computing. SAI 2022. Lecture Notes in Networks and Systems, vol 507. Springer, Cham. https://doi.org/10.1007/978-3-031-10464-0_10

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