Designs, Codes and Cryptography

, Volume 76, Issue 2, pp 269–277

# Permutation decoding of $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-linear codes

• José Joaquín Bernal
• Joaquim Borges
• Cristina Fernández-Córdoba
• Mercè Villanueva
Article

## Abstract

An alternative permutation decoding method is described which can be used for any binary systematic encoding scheme, regardless whether the code is linear or not. Thus, the method can be applied to some important codes such as $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-linear codes, which are binary and, in general, nonlinear codes in the usual sense. For this, it is proved that these codes allow a systematic encoding scheme. As particular examples, this permutation decoding method is applied to some Hadamard $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-linear codes.

## Keywords

Permutation decoding $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-linear codes Hadamard codes

94B60 94B25

## Notes

### Acknowledgments

The authors thank Prof. J. Rifà for valuable discussions in an earlier version of this paper. They also thank the anonymous referees for their valuable comments, which enabled them to improve the quality of the paper. This work was partially supported by the Spanish MICINN under Grants TIN2010-17358 and TIN2013-40524-P, and by the Catalan AGAUR under Grant 2009SGR1224. The authors are in alphabetical order.

## References

1. 1.
Assmus E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).Google Scholar
2. 2.
Borges J., Phelps K.T., Rifà J.: The rank and kernel of extended 1-perfect $$\mathbb{Z}_4$$-linear and additive non-$$\mathbb{Z}_4$$-linear codes. IEEE Trans. Inf. Theory 49(8), 2028–2034 (2003).Google Scholar
3. 3.
Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: $$\mathbb{Z}_2\mathbb{Z}_4$$-linear codes: generator matrices and duality. Des. Codes Cryptogr. 54, 167–179 (2010).Google Scholar
4. 4.
Cannon J.J., Bosma W.: Handbook of Magma Functions, 2.13th edn. North-Holland, Amsterdam (2006).Google Scholar
5. 5.
Fernández-Córdoba C., Pujol J., Villanueva M.: $$\mathbb{Z}_2\mathbb{Z}_4$$-linear codes: rank and kernel. Des. Codes Cryptogr. 56, 43–59 (2010).Google Scholar
6. 6.
Fish W., Key J.D., Mwambene, E.: Partial permutation decoding for simplex codes. Adv. Math. Commun. 6(4), 505–516 (2012).Google Scholar
7. 7.
Krotov D.S.: $$\mathbb{Z}_4$$-linear Hadamard and extended perfect codes. Electron. Notes Discret. Math. 6, 107–112 (2001).Google Scholar
8. 8.
MacWilliams F.J.: Permutation decoding of systematic codes. Bell Syst. Tech. J. 43, 485–505 (1964).Google Scholar
9. 9.
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).Google Scholar
10. 10.
Pernas J., Pujol J., Villanueva M.: Characterization of the automorphism group of quaternary linear Hadamard codes. Des. Codes Cryptogr. 70, 105–115 (2014).Google Scholar
11. 11.
Prange E.: The use of information sets in decoding cyclic codes. IEEE Trans. Inf. Theory 8(5), S5–S9 (1962).Google Scholar

## Authors and Affiliations

• José Joaquín Bernal
• 1
• Joaquim Borges
• 2
Email author
• Cristina Fernández-Córdoba
• 2
• Mercè Villanueva
• 2
1. 1.Department of MathematicsUniversidad de MurciaEspinardoSpain
2. 2.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaCerdanyola del VallèsSpain