One-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes

Abstract

In this paper, a class of additive codes which is referred to as \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes is introduced. This is a generalization towards another direction of recently introduced \(\mathbb {Z}_{2}\mathbb {Z}_{4}\) codes (Doughterty et al., Appl. Algebra Eng. Commun. Comput. 27(2), 123–138, 7). A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, the structure and possible weights for all one-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes are described. Additionally, we also construct some one-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes to illustrate our obtained results.

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Correspondence to Minjia Shi.

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This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

First Author is supported by the National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20) and Academic fund for outstanding talents in universities (gxbjZD03).

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Shi, M., Wang, C., Wu, R. et al. One-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes. Cryptogr. Commun. 12, 443–454 (2020). https://doi.org/10.1007/s12095-019-00391-5

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Keywords

  • Additive codes
  • One-weight codes
  • Two-weight codes
  • MacWilliams identity

Mathematics Subject Classification (2010)

  • 94B05
  • 94B15