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Designs, Codes and Cryptography

, Volume 83, Issue 1, pp 115–143 | Cite as

On linear codes admitting large automorphism groups

  • Nicola Pace
  • Angelo Sonnino
Article
  • 356 Downloads

Abstract

Linear codes with large automorphism groups are constructed. Most of them are suitable for permutation decoding. In some cases they are also optimal. For instance, we construct an optimal binary code of length \(n=252\) and dimension \(k=12\) having minimum distance \(d=120\) and automorphism group isomorphic to \(\text {PSL}(2,8)\rtimes \text {C}_{3}\).

Keywords

Linear code Projective space Automorphism Matrix group Permutation decoding 

Mathematics Subject Classification

51E22 20B25 94B35 

Notes

Acknowledgments

N. Pace was supported by the Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP), Proc. No. 12/03526-0. A. Sonnino was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) within the PRIN Project No. 2012XZE22K_006.

References

  1. 1.
    Abbott R., Bray J., Linton S., Nickerson S., Norton S., Parker R., Suleiman I., Tripp J., Walsh P., Wilson R.: Atlas of finite group representations—version 3. http://brauer.maths.qmul.ac.uk/Atlas/v3. Accessed 1 July 2015.
  2. 2.
    Berger T.P.: Cyclic alternant codes induced by an automorphism of a GRS code. In: Finite Fields: Theory, Applications, and Algorithms (Waterloo, ON, 1997), Contemporary Mathematics, vol. 225, pp. 143–154. American Mathematical Society, Providence (1999).Google Scholar
  3. 3.
    Bierbrauer J.: Introduction to Coding Theory, Discrete Mathematics and Its Applications. Chapman & Hall/CRC, Boca Raton (2005).Google Scholar
  4. 4.
    Bosma W., Cannon J.J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997).Google Scholar
  5. 5.
    Braun M., Kohnert A., Wassermann A.: Optimal linear codes from matrix groups. IEEE Trans. Inf. Theory 51(12), 4247–4251 (2005).Google Scholar
  6. 6.
    Camion P.: Linear codes with given automorphism groups. Discret. Math. 3, 33–45 (1972).Google Scholar
  7. 7.
    Cossidente A., Nolè C., Sonnino A.: Cap codes arising from duality. Bull. Inst. Combin. Appl. 67, 33–42 (2013).Google Scholar
  8. 8.
    Cossidente A., Sonnino A.: Finite geometry and the Gale transform. Discret. Math. 310(22), 3206–3210 (2010).Google Scholar
  9. 9.
    Cossidente A., Sonnino A.: Some recent results in finite geometry and coding theory arising from the Gale transform. Rend. Mat. Appl. (7) 30(1), 67–76. (2010).Google Scholar
  10. 10.
    Cossidente A., Sonnino A.: Linear codes arising from the Gale transform of distinguished subsets of some projective spaces. Discret. Math. 312(3), 647–651 (2012).Google Scholar
  11. 11.
    Crnković D., Rukavina S., Simčić L.: Binary doubly-even self-dual codes of length 72 with large automorphism groups. Math. Commun. 18(2), 297–308 (2013).Google Scholar
  12. 12.
    Fish W., Key J.D., Mwambene E.: Partial permutation decoding for simplex codes. Adv. Math. Commun. 6(4), 505–516 (2012).Google Scholar
  13. 13.
    Giulietti M., Korchmáros G., Marcugini S., Pambianco F.: Transitive \(A_6\)-invariant \(k\)-arcs in \(PG(2, q)\). Des. Codes Cryptogr. 68(1–3), 73–79 (2013).Google Scholar
  14. 14.
    Gordon D.M.: Minimal permutation sets for decoding the binary Golay codes. IEEE Trans. Inf. Theory 28(3), 541–543 (1982).Google Scholar
  15. 15.
    Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de. Accessed 2 Apr 2016.
  16. 16.
    Higman G.: On the simple group of D. G. Higman and C. C. Sims. Ill. J. Math. 13, 74–80 (1969).Google Scholar
  17. 17.
    Higman D.G., Sims C.C.: A simple group of order \(44,352,000\). Math. Z. 105, 110–113 (1968).Google Scholar
  18. 18.
    Hill R.: On the largest size of cap in \(S_{5,\,3}\), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 54(1973), 378–384 (1974).Google Scholar
  19. 19.
    Hill R.: A First Course in Coding Theory, Oxford Applied Mathematics and Computing Science Series. The Clarendon Press, Oxford University Press, New York (1986).Google Scholar
  20. 20.
    Hirschfeld J.W.P., Korchmaros G., Torres F.: Algebraic Curves Over a Finite Field. Princeton University Press, Princeton (2008).Google Scholar
  21. 21.
    Huffman W.C.: Codes and groups. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, part 2, vol. 2, pp. 1345–1440. North-Holland, Amsterdam (1998).Google Scholar
  22. 22.
    Indaco L., Korchmáros G.: 42-arcs in \(PG(2, q)\) left invariant by \(PSL(2,7)\). Des. Codes Cryptogr. 64(1–2), 33–46 (2012).Google Scholar
  23. 23.
    Key J.D.: Permutation decoding for codes from designs, finite geometries and graphs. In: Crnkovič D., Tonchev V. (eds.) Information Security, Coding Theory and Related Combinatorics. NATO Science for Peace and Security Series D: Information and Communication Security, vol. 29, pp. 172–201. IOS, Amsterdam (2011).Google Scholar
  24. 24.
    Key J.D., McDonough T.P., Mavron V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Comb. 26(5), 665–682 (2005).Google Scholar
  25. 25.
    Key J.D., Moori J., Rodrigues B.G.: Permutation decoding for the binary codes from triangular graphs. Eur. J. Comb. 25(1), 113–123 (2004).Google Scholar
  26. 26.
    Knapp W., Schaeffer H.-J.: On the codes related to the Higman–Sims graph. Electron. J. Comb. 22(1), P1–P19 (2015).Google Scholar
  27. 27.
    Knapp W., Schmid P.: Codes with prescribed permutation group. J. Algebra 67(2), 415–435 (1980).Google Scholar
  28. 28.
    Kohnert A.: Constructing two-weight codes with prescribed groups of automorphisms. Discret. Appl. Math. 155(11), 1451–1457 (2007).Google Scholar
  29. 29.
    Kohnert A., Wassermann A.: Construction of binary and ternary self-orthogonal linear codes. Discret. Appl. Math. 157(9), 2118–2123 (2009).Google Scholar
  30. 30.
    Kohnert A., Zwanzger J.: New linear codes with prescribed group of automorphisms found by heuristic search. Adv. Math. Commun. 3(2), 157–166 (2009).Google Scholar
  31. 31.
    Korchmáros G., Pace N.: Infinite family of large complete arcs in \(\text{ PG }(2, q^n)\), with \(q\) odd and \(n>1\) odd. Des. Codes Cryptogr. 55(2–3), 285–296 (2010).Google Scholar
  32. 32.
    Kramer E.S., Mesner D.M.: \(t\)-Designs on hypergraphs. Discret. Math. 15, 263–296 (1976).Google Scholar
  33. 33.
    Kroll H.-J., Vincenti R.: PD-sets for the codes related to some classical varieties. Discret. Math. 301(1), 89–105 (2005).Google Scholar
  34. 34.
    Kroll H.-J., Vincenti R.: Antiblocking systems and PD-sets. Discret. Math. 308(2–3), 401–407 (2008).Google Scholar
  35. 35.
    Kroll H.-J., Vincenti R.: PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of PG(5, 2). Discret. Math. 308(2–3), 408–414 (2008).Google Scholar
  36. 36.
    MacWilliams F.J.: Permutation decoding of systematic codes. Bell System Tech. J. 43(1), 485–505 (1964).Google Scholar
  37. 37.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. I. North-Holland Mathematical Library, vol. 16. North-Holland, Amsterdam (1977).Google Scholar
  38. 38.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. II. North-Holland Mathematical Library, vol. 16. North-Holland, Amsterdam (1977).Google Scholar
  39. 39.
    Martis M., Bamberg J., Morris S.: An enumeration of certain projective ternary two-weight codes. J. Comb. Des. 24(1), 21–35 (2016).Google Scholar
  40. 40.
    Pace N.: New ternary linear codes from projectivity groups. Discret. Math. 331, 22–26 (2014).Google Scholar
  41. 41.
    Pace N.: On small complete arcs and transitive \(A_5\)-invariant arcs in the projective plane \(PG(2, q)\). J. Comb. Des. 22(10), 425–434 (2014).Google Scholar
  42. 42.
    Rodrigues B.G.: Self-orthogonal designs and codes from the symplectic groups \(S_4(3)\) and \(S_4(4)\). Discret. Math. 308(10), 1941–1950 (2008).Google Scholar
  43. 43.
    Sonnino A.: Transitive PSL(2, 7)-invariant 42-arcs in 3-dimensional projective spaces. Des. Codes Cryptogr. 72(2), 455–463 (2014).Google Scholar
  44. 44.
    Tolhuizen L.M.G.M., van Gils W.J.: A large automorphism group decreases the number of computations in the construction of an optimal encoder/decoder pair for a linear block code. IEEE Trans. Inf. Theory 34(2), 333–338 (1988).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesDublin Institute of TechnologyDublin 8Ireland
  2. 2.Dipartimento di Matematica, Informatica ed EconomiaUniversità degli Studi della BasilicataPotenzaItaly

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