Skip to main content
Top
Published in: Designs, Codes and Cryptography 2/2014

01-08-2014

\(\mathbb{Z }_2\mathbb{Z }_4\)-Additive formally self-dual codes

Authors: S. T. Dougherty, C. Fernández-Córdoba

Published in: Designs, Codes and Cryptography | Issue 2/2014

Login to get access

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We study odd and even \(\mathbb{Z }_2\mathbb{Z }_4\) formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.
Literature
1.
go back to reference Bannai E., Dougherty S.T., Harada M., Oura M.: Type II codes, even unimodular lattices, and invariant rings. IEEE Trans. Inf. Theory 45(4), 1194–1205 (1999). Bannai E., Dougherty S.T., Harada M., Oura M.: Type II codes, even unimodular lattices, and invariant rings. IEEE Trans. Inf. Theory 45(4), 1194–1205 (1999).
2.
go back to reference Bierbrauer J.: Introduction to Coding Theory. Chapman & Hall, New York (2005). Bierbrauer J.: Introduction to Coding Theory. Chapman & Hall, New York (2005).
3.
go back to reference Blokhuis A., Brouwer A.E.: Small additive quaternary codes. Eur. J. Comb. 25, 161–167 (2004). Blokhuis A., Brouwer A.E.: Small additive quaternary codes. Eur. J. Comb. 25, 161–167 (2004).
4.
go back to reference Borges J., Rifà J.: A characterization of 1-perfect additive codes. IEEE Trans. Inf. Theory 45(5), 1688–1697 (1999). Borges J., Rifà J.: A characterization of 1-perfect additive codes. IEEE Trans. Inf. Theory 45(5), 1688–1697 (1999).
5.
go back to reference Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: On \(\mathbb{Z}_2 \mathbb{Z}_4\)-linear codes and duality. In: V Jornades de Matemàtica Discreta i Algorísmica, Soria (Spain). 11–14, Jul., pp. 171–177 (2006). Borges J., Fernández-Córdoba C., Pujol J., Rifà J., Villanueva M.: On \(\mathbb{Z}_2 \mathbb{Z}_4\)-linear codes and duality. In: V Jornades de Matemàtica Discreta i Algorísmica, Soria (Spain). 11–14, Jul., pp. 171–177 (2006).
6.
go back to reference 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(2), 167–179 (2010). 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(2), 167–179 (2010).
7.
go back to reference Borges J., Dougherty S.T., Fernández-Córdoba C.: Characterization and constructions of self-dual codes over \(\mathbb{Z}_2 \times \mathbb{Z}_4\). Adv. Math. Commun. 6(3), 287–303 (2012). Borges J., Dougherty S.T., Fernández-Córdoba C.: Characterization and constructions of self-dual codes over \(\mathbb{Z}_2 \times \mathbb{Z}_4\). Adv. Math. Commun. 6(3), 287–303 (2012).
8.
go back to reference Broué M., Enguehard M.: Polynômes des poids de certains codes et fonctions thêta de certains réseaux”. Ann. Sci. École Norm. Sup. 5(4), 157–181 (1972). Broué M., Enguehard M.: Polynômes des poids de certains codes et fonctions thêta de certains réseaux”. Ann. Sci. École Norm. Sup. 5(4), 157–181 (1972).
9.
go back to reference Brualdi R.A., Pless V.S.: Weight enumerators of self-dual codes. IEEE Trans. Inf. Theory IT-37, 1222–1225 (1991). Brualdi R.A., Pless V.S.: Weight enumerators of self-dual codes. IEEE Trans. Inf. Theory IT-37, 1222–1225 (1991).
10.
go back to reference Conway J.H., Sloane N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inf. Theory 36(6), 1319–1333 (1990). Conway J.H., Sloane N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inf. Theory 36(6), 1319–1333 (1990).
11.
go back to reference Conway J.H., Sloane N.J.A.: Self-dual codes over the integers modulo 4. J. Comb. Theory Ser. A 62, 30–45 (1993). Conway J.H., Sloane N.J.A.: Self-dual codes over the integers modulo 4. J. Comb. Theory Ser. A 62, 30–45 (1993).
12.
go back to reference Conway J.H., Sloane N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, Berlin (1999). Conway J.H., Sloane N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, Berlin (1999).
13.
go back to reference Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973). Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973).
14.
go back to reference Delsarte P., Levenshtein V.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44(6), 2477–2504 (1998). Delsarte P., Levenshtein V.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44(6), 2477–2504 (1998).
15.
go back to reference Dougherty S.T.: Shadow codes and their weight enumerators. IEEE Trans. Inf. Theory 41(3) (1995). Dougherty S.T.: Shadow codes and their weight enumerators. IEEE Trans. Inf. Theory 41(3) (1995).
16.
go back to reference Dougherty S.T.: Formally self-dual codes and gray maps. In: Proceedings of ACCT2012, Pomorie Bulgaria (2012). Dougherty S.T.: Formally self-dual codes and gray maps. In: Proceedings of ACCT2012, Pomorie Bulgaria (2012).
17.
go back to reference Dougherty S.T., Fernandez-Cordoba C.: Codes over \(\mathbb{Z}_{2^k}\), gray maps and self-dual codes. Adv. Math. Commun. 5(4), 571–588 (2011). Dougherty S.T., Fernandez-Cordoba C.: Codes over \(\mathbb{Z}_{2^k}\), gray maps and self-dual codes. Adv. Math. Commun. 5(4), 571–588 (2011).
18.
go back to reference Dougherty S.T., Solé P.: Shadow constructions of unimodular lattices and self-dual codes. In: Sunada T., Sy B.W., Lo Y. (eds.) AMC, pp. 139–152. World Scientific, Singapore (2002). Dougherty S.T., Solé P.: Shadow constructions of unimodular lattices and self-dual codes. In: Sunada T., Sy B.W., Lo Y. (eds.) AMC, pp. 139–152. World Scientific, Singapore (2002).
19.
go back to reference Dougherty S.T., Kim J.-L., Liu H.: Constructions of self-dual codes over finite commutative chain rings. Int. J. Inf. Coding Theory 1(2), 171–190 (2010). Dougherty S.T., Kim J.-L., Liu H.: Constructions of self-dual codes over finite commutative chain rings. Int. J. Inf. Coding Theory 1(2), 171–190 (2010).
20.
go back to reference Dougherty S.T., Kim J.-L., Kulosman H., Liu H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16(1), 14–26 (2010). Dougherty S.T., Kim J.-L., Kulosman H., Liu H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16(1), 14–26 (2010).
21.
go back to reference Fernández-Córdoba C., Pujol J., Villanueva M.: \(\mathbb{Z}_2\mathbb{Z}_4\)-linear codes: rank and kernel. Des. Codes Cryptogr. 56(1), 43–59 (2010). Fernández-Córdoba C., Pujol J., Villanueva M.: \(\mathbb{Z}_2\mathbb{Z}_4\)-linear codes: rank and kernel. Des. Codes Cryptogr. 56(1), 43–59 (2010).
22.
go back to reference Gleason A. M.: Weight polynomials of self-dual codes and the MacWilliams identities. Actes du Congrs International des Mathmaticiens (Nice, 1970), Tome 3, pp. 211–215. Gauthier-Villars, Paris (1971). Gleason A. M.: Weight polynomials of self-dual codes and the MacWilliams identities. Actes du Congrs International des Mathmaticiens (Nice, 1970), Tome 3, pp. 211–215. Gauthier-Villars, Paris (1971).
23.
go back to reference Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(\mathbb{Z}_4\)-linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994). Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(\mathbb{Z}_4\)-linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994).
24.
go back to reference Han S., Lee H., Lee Y.: Binary Formally Self-Dual Odd Codes. Des. Codes Cryptogr. 61, 141–150 (2011). Han S., Lee H., Lee Y.: Binary Formally Self-Dual Odd Codes. Des. Codes Cryptogr. 61, 141–150 (2011).
25.
go back to reference Kim J.-L., Pless V.: Designs in additive codes over GF(4). Des. Codes Cryptogr. 30, 187–199 (2003). Kim J.-L., Pless V.: Designs in additive codes over GF(4). Des. Codes Cryptogr. 30, 187–199 (2003).
26.
go back to reference MacWilliams F.J., Sloane N.J.A.: The Theory of Error-correcting Codes. North-Holland Mathematical Library, vol. 16. North-Holland, Amsterdam (1977). MacWilliams F.J., Sloane N.J.A.: The Theory of Error-correcting Codes. North-Holland Mathematical Library, vol. 16. North-Holland, Amsterdam (1977).
27.
go back to reference Odlyzko A.M., Sloane N.J.A.: A theta-function identity for nonlattice packings. Stud. Sci. Math. Hungarica 5, 461–465 (1980). Odlyzko A.M., Sloane N.J.A.: A theta-function identity for nonlattice packings. Stud. Sci. Math. Hungarica 5, 461–465 (1980).
28.
go back to reference Phelps K.T., Rifà J.: On binary 1-perfect additive codes: some structural properties. IEEE Trans. Inf. Theory 48(9), 2587–2592 (2002). Phelps K.T., Rifà J.: On binary 1-perfect additive codes: some structural properties. IEEE Trans. Inf. Theory 48(9), 2587–2592 (2002).
29.
go back to reference Pujol J., Rifà J.: Translation invariant propelinear codes. IEEE Trans. Inf. Theory 43, 590–598 (1997). Pujol J., Rifà J.: Translation invariant propelinear codes. IEEE Trans. Inf. Theory 43, 590–598 (1997).
30.
go back to reference Rains E., Sloane N.J.A.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds.) The Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998). Rains E., Sloane N.J.A.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds.) The Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998).
31.
go back to reference Ward H.N.: A restriction on the weight enumerator of a self-dual code. J. Comb. Theory Ser. A 21, 253–255 (1976). Ward H.N.: A restriction on the weight enumerator of a self-dual code. J. Comb. Theory Ser. A 21, 253–255 (1976).
Metadata
Title
-Additive formally self-dual codes
Authors
S. T. Dougherty
C. Fernández-Córdoba
Publication date
01-08-2014
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 2/2014
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-012-9773-4

Other articles of this Issue 2/2014

Designs, Codes and Cryptography 2/2014 Go to the issue

Premium Partner