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On two-weight \(\mathbb {Z}_{2^k}\)-codes. (English) Zbl 1387.94118

Summary: 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.

MSC:

94B05 Linear codes (general theory)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05E30 Association schemes, strongly regular graphs
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[1] Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2011). · Zbl 1231.05001
[2] Byrne E., Greferath M., Honold T.: Ring geometries, two-weight codes and strongly regular graphs. Des. Codes Cryptogr. 48, 1-16 (2008). · Zbl 1184.94266
[3] Byrne E., Kiermaier M., Sneyd A.: Properties of codes with two homogeneous weights. Finite Fields Appl. 18, 711-727 (2012). · Zbl 1245.05135
[4] Calderbank R.: On uniformly packed \[[n,n-k,4]\][n,n-k,4] codes over \[GF(q)\] GF(q) and a class of caps in \[PG(k-1,q)\] PG(k-1,q). J. Lond. Math. Soc. (2) 26, 365-384 (1982). · Zbl 0545.94014
[5] Carlet \[C.: {\mathbb{Z}}_{2^k}\] Z2k-linear codes. IEEE Trans. Inf. Theory 44, 1543-1547 (1998). · Zbl 0935.94028
[6] Constantinescu I., Heise W.: A metric for codes over residue class rings of integers. Probl. Inf. Transm. 33, 208-213 (1997). · Zbl 0977.94055
[7] Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3, 47-64 (1972). · Zbl 0245.94010
[8] Delsarte P.: An algebraic approach to the association schemes of Coding Theory. Philips Research Reports Supplement No. 10 (1973) · Zbl 1075.05606
[9] Hammons R., Kumar V.P., Calderbank A.R., Sloane N.J.A., Solé P.: The \[{\mathbb{Z}}_4-\] Z4-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40, 301-319 (1994). · Zbl 0811.94039
[10] Honold T.: A characterization of finite Frobenius rings. Arch. Math. 76, 406-415 (2001). · Zbl 0984.16017
[11] Honold T.: Two-intersection sets in projective Hjemslev spaces. In: Edelmayer A. (ed.) Proceedings of MTNS 2010, Budapest, pp. 1807-1813 (2010) · Zbl 1317.94163
[12] Krotov D.: \[{\mathbb{Z}}_4-\] Z4-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) · Zbl 0987.94513
[13] Krotov D.: On \[{\mathbb{Z}}_{2^k}-\] Z2k-dual binary codes. IEEE Trans. Inf. Theory 53, 1532-1537 (2007). · Zbl 1205.94121
[14] Ling S., Xing C.P.: Coding Theory: A First Course. Cambridge University Press, New York (2004).
[15] Phelps K.T., Rifà J., Villanueva M.: On the additive \[({\mathbb{Z}}_4-\](Z4-linear and non-\[{\mathbb{Z}}_4-\] Z4-linear ) Hadamard codes. Rank and Kernel. IEEE Trans. Inf. Theory 52, 316-319 (2006). · Zbl 1317.94163
[16] Shi M.J., Wang Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over \[{\mathbb{Z}}_4,\] Z4,. J. Syst. Sci. Complex. 27, 795-810 (2014). · Zbl 1316.94111
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