# Codes and the Steenrod algebra

Keywords:
Self-dual codes, Non-commutative rings, Steenrod algebra

### 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

A. R. Calderbank, N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptog. 6(1) (1995) 21–35.

Y. J. Choie, S. T. Dougherty, Codes over Sigma_{2m}Σ

2m

and Jacobi forms over the quaternions, Appl. Algebra Engrg. Comm. Comput. 15(2) (2004) 129–147.

Y. J. Choie, S. T. Dougherty, Codes over rings, complex lattices and Hermitian modular forms, European J. Combin. 26(2) (2005) 145–165.

S. T. Dougherty, A. Leroy, Euclidean self–dual codes over non–commuatative Frobenius rings, Appl. Alg. Engrg. Comm. Comp. 27 (3) (2016) 185–203.

S. T. Dougherty, Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptog. 39(1) (2006) 65–80.

A. Kruckman, https://math.berkeley.edu/kruckman/adem/.

J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67(1) (1958) 150–171.

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.

J. P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–Mac–Lane, Comment. Math. Helv. 27(1) (1953) 198–232.

N. E. Steenrod, D. B. A. Epstein, Cohomology Operations, Ann. of Math. Studies, no.50, Princeton University Press, 1962.

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.

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.

R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. Lond. Math. Soc. 27(4) (1995) 380–386.

R. M. W. Wood, Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30(5) (1998) 449–517.

Y. J. Choie, S. T. Dougherty, Codes over Sigma_{2m}Σ

2m

and Jacobi forms over the quaternions, Appl. Algebra Engrg. Comm. Comput. 15(2) (2004) 129–147.

Y. J. Choie, S. T. Dougherty, Codes over rings, complex lattices and Hermitian modular forms, European J. Combin. 26(2) (2005) 145–165.

S. T. Dougherty, A. Leroy, Euclidean self–dual codes over non–commuatative Frobenius rings, Appl. Alg. Engrg. Comm. Comp. 27 (3) (2016) 185–203.

S. T. Dougherty, Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptog. 39(1) (2006) 65–80.

A. Kruckman, https://math.berkeley.edu/kruckman/adem/.

J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67(1) (1958) 150–171.

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.

J. P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–Mac–Lane, Comment. Math. Helv. 27(1) (1953) 198–232.

N. E. Steenrod, D. B. A. Epstein, Cohomology Operations, Ann. of Math. Studies, no.50, Princeton University Press, 1962.

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.

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.

R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. Lond. Math. Soc. 27(4) (1995) 380–386.

R. M. W. Wood, Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30(5) (1998) 449–517.

Published

2017-05-15

Section

Articles