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Codes and the Steenrod algebra

Year 2017, , 141 - 154, 10.01.2017
https://doi.org/10.13069/jacodesmath.284950

Abstract

We study codes over the finite sub Hopf algebras of the Steenrod algebra. We define three dualities
for codes over these rings, namely the Eulidean duality, the Hermitian duality and a duality based
on the underlying additive group structure. We study self-dual codes, namely codes equal to their
orthogonal, with respect to all three dualities.

References

  • [1] A. R. Calderbank, N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptog. 6(1) (1995) 21–35.
  • [2] Y. J. Choie, S. T. Dougherty, Codes over $\Sigma_{2m}$ and Jacobi forms over the quaternions, Appl. Algebra Engrg. Comm. Comput. 15(2) (2004) 129–147.
  • [3] Y. J. Choie, S. T. Dougherty, Codes over rings, complex lattices and Hermitian modular forms, European J. Combin. 26(2) (2005) 145–165.
  • [4] S. T. Dougherty, A. Leroy, Euclidean self–dual codes over non–commuatative Frobenius rings, Appl. Alg. Engrg. Comm. Comp. 27 (3) (2016) 185–203.
  • [5] S. T. Dougherty, Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptog. 39(1) (2006) 65–80.
  • [6] A. Kruckman, https://math.berkeley.edu/kruckman/adem/.
  • [7] J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67(1) (1958) 150–171.
  • [8] G. Nebe, E. M. Rains, N. J. A. Sloane, Self–Dual Codes and Invariant Theory, Vol. 17, Algorithms and Computation in Mathematics, Springer–Verlag, Berlin, 2006.
  • [9] J. P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–Mac–Lane, Comment. Math. Helv. 27(1) (1953) 198–232.
  • [10] N. E. Steenrod, D. B. A. Epstein, Cohomology Operations, Ann. of Math. Studies, no.50, Princeton University Press, 1962.
  • [11] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.
  • [12] J. A.Wood, Anti–isomorphisms, character modules, and self–dual codes over non–commutative rings, Int. J. Inf. Coding Theory 1(4) (2010) 429–444.
  • [13] R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. Lond. Math. Soc. 27(4) (1995) 380–386.
  • [14] R. M. W. Wood, Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30(5) (1998) 449–517.
Year 2017, , 141 - 154, 10.01.2017
https://doi.org/10.13069/jacodesmath.284950

Abstract

References

  • [1] A. R. Calderbank, N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptog. 6(1) (1995) 21–35.
  • [2] Y. J. Choie, S. T. Dougherty, Codes over $\Sigma_{2m}$ and Jacobi forms over the quaternions, Appl. Algebra Engrg. Comm. Comput. 15(2) (2004) 129–147.
  • [3] Y. J. Choie, S. T. Dougherty, Codes over rings, complex lattices and Hermitian modular forms, European J. Combin. 26(2) (2005) 145–165.
  • [4] S. T. Dougherty, A. Leroy, Euclidean self–dual codes over non–commuatative Frobenius rings, Appl. Alg. Engrg. Comm. Comp. 27 (3) (2016) 185–203.
  • [5] S. T. Dougherty, Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptog. 39(1) (2006) 65–80.
  • [6] A. Kruckman, https://math.berkeley.edu/kruckman/adem/.
  • [7] J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67(1) (1958) 150–171.
  • [8] G. Nebe, E. M. Rains, N. J. A. Sloane, Self–Dual Codes and Invariant Theory, Vol. 17, Algorithms and Computation in Mathematics, Springer–Verlag, Berlin, 2006.
  • [9] J. P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–Mac–Lane, Comment. Math. Helv. 27(1) (1953) 198–232.
  • [10] N. E. Steenrod, D. B. A. Epstein, Cohomology Operations, Ann. of Math. Studies, no.50, Princeton University Press, 1962.
  • [11] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.
  • [12] J. A.Wood, Anti–isomorphisms, character modules, and self–dual codes over non–commutative rings, Int. J. Inf. Coding Theory 1(4) (2010) 429–444.
  • [13] R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. Lond. Math. Soc. 27(4) (1995) 380–386.
  • [14] R. M. W. Wood, Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30(5) (1998) 449–517.
There are 14 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Steven T. Dougherty This is me

Tane Vergili

Publication Date January 10, 2017
Published in Issue Year 2017

Cite

APA Dougherty, S. T., & Vergili, T. (2017). Codes and the Steenrod algebra. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2 (Special Issue: Noncommutative rings and their applications), 141-154. https://doi.org/10.13069/jacodesmath.284950
AMA Dougherty ST, Vergili T. Codes and the Steenrod algebra. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2017;4(2 (Special Issue: Noncommutative rings and their applications):141-154. doi:10.13069/jacodesmath.284950
Chicago Dougherty, Steven T., and Tane Vergili. “Codes and the Steenrod Algebra”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 2 (Special Issue: Noncommutative rings and their applications) (May 2017): 141-54. https://doi.org/10.13069/jacodesmath.284950.
EndNote Dougherty ST, Vergili T (May 1, 2017) Codes and the Steenrod algebra. Journal of Algebra Combinatorics Discrete Structures and Applications 4 2 (Special Issue: Noncommutative rings and their applications) 141–154.
IEEE S. T. Dougherty and T. Vergili, “Codes and the Steenrod algebra”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), pp. 141–154, 2017, doi: 10.13069/jacodesmath.284950.
ISNAD Dougherty, Steven T. - Vergili, Tane. “Codes and the Steenrod Algebra”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/2 (Special Issue: Noncommutative rings and their applications) (May 2017), 141-154. https://doi.org/10.13069/jacodesmath.284950.
JAMA Dougherty ST, Vergili T. Codes and the Steenrod algebra. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:141–154.
MLA Dougherty, Steven T. and Tane Vergili. “Codes and the Steenrod Algebra”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), 2017, pp. 141-54, doi:10.13069/jacodesmath.284950.
Vancouver Dougherty ST, Vergili T. Codes and the Steenrod algebra. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(2 (Special Issue: Noncommutative rings and their applications):141-54.