Abstract
We determine the possible homogeneous weights of regular projective two-weight codes over \(\mathbb {Z}_{2^k}\) of length \(n>3\), with dual Krotov distance \(d^{\lozenge }\) at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When \(k=2\), we characterize the parameters of such codes as those of the inverse Gray images of \(\mathbb {Z}_4\)-linear Hadamard codes, which have been characterized by their types by several authors.
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Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2011).
Byrne E., Greferath M., Honold T.: Ring geometries, two-weight codes and strongly regular graphs. Des. Codes Cryptogr. 48, 1–16 (2008).
Byrne E., Kiermaier M., Sneyd A.: Properties of codes with two homogeneous weights. Finite Fields Appl. 18, 711–727 (2012).
Calderbank R.: On uniformly packed \([n,n-k,4]\) codes over \(GF(q)\) and a class of caps in \(PG(k-1,q)\). J. Lond. Math. Soc. (2) 26, 365–384 (1982).
Carlet C.: \({\mathbb{Z}}_{2^k}\)-linear codes. IEEE Trans. Inf. Theory 44, 1543–1547 (1998).
Constantinescu I., Heise W.: A metric for codes over residue class rings of integers. Probl. Inf. Transm. 33, 208–213 (1997).
Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3, 47–64 (1972).
Delsarte P.: An algebraic approach to the association schemes of Coding Theory. Philips Research Reports Supplement No. 10 (1973)
Hammons R., Kumar V.P., 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, 301–319 (1994).
Honold T.: A characterization of finite Frobenius rings. Arch. Math. 76, 406–415 (2001).
Honold T.: Two-intersection sets in projective Hjemslev spaces. In: Edelmayer A. (ed.) Proceedings of MTNS 2010, Budapest, pp. 1807–1813 (2010)
Krotov D.: \({\mathbb{Z}}_4-\)linear Hadamard and extended perfect codes. In: Proceedings of the International Workshop on Coding and Cryptography WCC 2001, Paris, January 2001, pp. 329–334. Electronic Notes in Discrete Mathematics, vol. 6, pp. 107–112 (2001)
Krotov D.: On \({\mathbb{Z}}_{2^k}-\)dual binary codes. IEEE Trans. Inf. Theory 53, 1532–1537 (2007).
Ling S., Xing C.P.: Coding Theory: A First Course. Cambridge University Press, New York (2004).
Phelps K.T., Rifà J., Villanueva M.: On the additive \(({\mathbb{Z}}_4-\)linear and non-\({\mathbb{Z}}_4-\)linear ) Hadamard codes. Rank and Kernel. IEEE Trans. Inf. Theory 52, 316–319 (2006).
Shi M.J., Wang Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over \({\mathbb{Z}}_4,\). J. Syst. Sci. Complex. 27, 795–810 (2014).
Acknowledgements
The authors are grateful to the anonymous referees for helpful remarks. This research is supported by National Natural Science Foundation of China (61672036), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).
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Communicated by J. H. Koolen.
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Shi, M., Sepasdar, Z., Alahmadi, A. et al. On two-weight \(\mathbb {Z}_{2^k}\)-codes. Des. Codes Cryptogr. 86, 1201–1209 (2018). https://doi.org/10.1007/s10623-017-0390-0
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DOI: https://doi.org/10.1007/s10623-017-0390-0