main-content

## Swipe to navigate through the articles of this issue

Published in:

27-04-2020 | Original Paper

# Self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 4n

Authors: Yuan Cao, Yonglin Cao, Fang-Wei Fu, Guidong Wang

## Abstract

For any odd positive integer n, we express cyclic codes over $${\mathbb {Z}}_4$$ of length 4n in a new way. Based on the expression of each cyclic code $${\mathcal {C}}$$, we provide an efficient encoder and determine the type of $${\mathcal {C}}$$. In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 4n and correct a mistake in the paper “Concatenated structure of cyclic codes over $${\mathbb {Z}}_4$$ of length 4n” (Cao et al. in Appl Algebra Eng Commun Comput 10:279–302, 2016). In addition, we obtain 50 new self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 28.
Literature
1.
Abualrub, T., Oehmke, R.: On the generators of $${\mathbb{Z}}_4$$ cyclic codes of length $$2^e$$. IEEE Trans. Inform. Theory 49, 2126–2133 (2003)
2.
Blackford, T.: Cyclic codes over $${\mathbb{Z}}_4$$ of oddly even length. Discrete Appl. Math. 128, 27–46 (2003)
3.
Calderbank, A.R., Sloane, N.J.A.: Modular and $$p$$-adic cyclic codes. Des. Codes Cryptogr. 6, 21–35 (1995)
4.
Calderbank, A.R., Sloane, N.J.A.: Double circulant codes over $${\mathbb{Z}}_4$$ and even unimodular lattices. J. Algebraic Combin. 6, 119–131 (1997)
5.
Cao, Y., Cao, Y., Li, Q.: Concatenated structure of cyclic codes over $${\mathbb{Z}}_4$$ of length $$4n$$. Appl. Algebra Eng. Commun. Comput. 10, 279–302 (2016) CrossRef
6.
Cao, Y., Cao, Y., Dougherty, S.T., Ling, S.: Construction and enumeration for self-dual cyclic codes over $${\mathbb{Z}}_4$$ of oddly even length. Des. Codes Cryptogr. 87, 2419–2446 (2019)
7.
Cao, Y.: A class of 1-generator repeated root quasi-cyclic codes. Des. Codes Cryptogr. 72, 483–496 (2014)
8.
Cao, Y., Cao, Y., Fu, F.-W.: On self-duality and hulls of cyclic codes over $$\frac{{\mathbb{F}}_{2^m}[u]}{\langle u^k\rangle }$$ with oddly even length. Appl. Algebra Eng. Commun. Comput. (2019). https://​doi.​org/​10.​1007/​s00200-019-00408-9
9.
Cao, Y., Cao, Y.: Negacyclic codes over the local ring $${\mathbb{Z}}_4[v]/\langle v^2+2v\rangle$$ of oddly even length and their Gray images. Finite Fields Appl. 52, 67–93 (2018)
10.
Cao, Y., Cao, Y.: Complete classification for simple root cyclic codes over the local ring $${\mathbb{Z}}_4[v]/\langle v^2+2v\rangle$$. Cryptogr. Commun. 12, 301–319 (2020)
11.
Dougherty, S.T., Ling, S.: Cyclic codes over $${\mathbb{Z}}_4$$ of even length. Des. Codes Cryptogr. 39, 127–153 (2006)
12.
Gaborit, P., Natividad, A.M., Solé, P.: Eisenstein lattices, Galois rings and quaternary codes. Int. J. Number Theory 2, 289–303 (2006)
13.
Hammons Jr., 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. Inform. Theory 40(2), 301–319 (1994)
14.
Harada, M.: Self-dual $${\mathbb{Z}}_4$$-codes and Hadamard matrices. Discrete Math. 245, 273–278 (2002)
15.
Harada, M., Kitazume, M., Munemasa, A., Venkov, B.: On some self-dual codes and unimodular lattices in dimension 48. Eur. J. Combin. 26, 543–557 (2005)
16.
Harada, M., Miezaki, T.: An optimal odd unimodular lattice in dimension 72. Arch. Math. 97(6), 529–533 (2011)
17.
Harada M., Solé P., Gaborit P.: Self-dual codes over $${\mathbb{Z}}_4$$ and unimodular lattices: a survey. In: Algebras and Combinatorics, Hong Kong, 1997, pp. 255–275. Springer, Singapore (1999)
18.
Kiah, H.M., Leung, K.H., Ling, S.: A note on cyclic codes over $${{\rm GR}}(p^2, m)$$ of length $$p^k$$. Des. Codes Cryptogr. 63, 105–112 (2012)
19.
Jitman, S., Ling, S., Sangwisut, E.: On self-dual cyclic codes of length $$p^a$$ over $${{\rm GR}}(p^2, s)$$. Adv. Math. Commun. 10, 255–273 (2016)
20.
Pless, V.S., Qian, Z.: Cyclic codes and quadratic residue codes over $${\mathbb{Z}}_4$$. IEEE Trans. Inform. Theory 42, 1594–1600 (1996)
21.
Pless, V.S., Solé, P., Qian, Z.: Cyclic self-dual $${\mathbb{Z}}_4$$-codes. Finite Fields Appl. 3, 48–69 (1997)
22.
Shi, M., Qian, L., Sok, L., Aydin, N., Solé, P.: On constacyclic codes over $${\mathbb{Z}}_4[u]/\langle u^2-1\rangle$$ and their Gray images. Finite Fields Appl. 45, 86–95 (2017)
23.
Wan, Z.-X.: Quaternary Codes. World Scientific Pub Co Inc., Singapore (1997) CrossRef
24.
Wan, Z.-X.: Lectures on Finite Fields and Galois Rings. World Scientific Pub Co Inc., Singapore (2003) CrossRef
25.
Database of $${\mathbb{Z}}_4$$ codes [online], http://​www.​z4codes.​info. Accessed 03 Sept 2016
Title
Self-dual cyclic codes over of length 4n
Authors
Yuan Cao
Yonglin Cao
Fang-Wei Fu
Guidong Wang
Publication date
27-04-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 1/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00424-0

Go to the issue