Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 3/2016

01-03-2016

Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

Authors: Maosheng Xiong, Nian Li, Zhengchun Zhou, Cunsheng Ding

Published in: Designs, Codes and Cryptography | Issue 3/2016

Login to get access

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

search-config
loading …

Abstract

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279–3286, 2010; IEEE Trans Inf Theory 60:3903–3912, 2014), we study two families of cyclic codes over \({\mathbb F}_p\) with arbitrary number of zeroes of generalized Niho type, more precisely \({\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}\) (for \(p=2\)) of \(t+1\) zeroes, and \({\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}\) (for any prime \(p\)) of \(t\) zeroes for any \(t\). We find that the first family has at most \((2t+1)\) non-zero weights, and the second has at most \(2t\) non-zero weights. Their weight distribution are also determined in the paper.
previous article Weight enumerator of some irreducible cyclic codes
next article The weight distribution of a family of -ary cyclic codes
Appendix
Available only for authorised users
Literature
1.
Aubry Y., Langevin P.: On the weights of binary irreducible cyclic codes. In: Proceedings of the 2005 International Conference on Coding and Cryptography. Lecture Notes in Computer Science, vol. 3969, pp. 46–54. Springer, Berlin (2006).
2.
Baumert L.D., McEliece R.J.: Weights of irreducible cyclic codes. Inf. Control 20, 158–175 (1972).
3.
Baumert L.D., Mykkeltveit J.: Weight distributions of some irreducible cyclic codes. DSN Prog. Rep. 16, 128–131 (1973).
4.
Calderbankand R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).
5.
Charpin P.: Cyclic codes with few weights and Niho exponents. J. Comb. Theory Ser. A 108, 247–259 (2004).
6.
Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313, 434–446 (2013).
7.
Ding C., Yang Y., Tang X.: Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inf. Theory 56, 3605–3612 (2010).
8.
Ding C., Liu Y., Ma C., Zeng L.: The weight distributions of the duals of cyclic codes with two zeroes. IEEE Trans. Inf. Theory 57, 8000–8006 (2011).
9.
Feng T.: On cyclic codes of length \(2^{2^r}-1\) with two zeroes whose dual codes have three weights. Des. Codes Cryptogr. 62, 253–258 (2012).
10.
Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14, 390–409 (2008).
11.
Feng T., Momihara K.: Evaluation of the weight distribution of a class of cyclic codes based on index 2 Gauss sums. IEEE Trans. Inf. Theory 59, 5980–5984 (2013).
12.
Fitzgerald R., Yucas J.: Sums of Gauss sums and weights of irreducible codes. Finite Fields Appl. 11, 89–110 (2005).
13.
Hollmann H.D.L., Xiang Q.: On binary cyclic codes with few weights. In: Proceedings of the Finite Fields and Applications (Augsburg), Berlin, pp. 251–275 (1999).
14.
Kløve T.: Codes for Error Detection. World Scientific, Singapore (2007).
15.
Li C., Zeng X., Hu L.: A class of binary cyclic codes with five weights. Sci. China Math. 53, 3279–3286 (2010).
16.
Li N., Helleseth T., Kholosha A., Tang X.: On the Walsh transform of a class of functions from Niho exponents. IEEE Trans. Inf. Theory 59, 4662–4667 (2013).
17.
Li S., Hu S., Feng T., Ge G.: The weight distribution of a class of cyclic codes related to Hermitian forms graphs. IEEE Trans. Inf. Theory 59, 3064–3067 (2013).
18.
Li S., Feng T., Ge G.: On the weight distribution of cyclic codes with Niho exponents. IEEE Trans. Inf. Theory 60, 3903–3912 (2014).
19.
Luo J., Feng K.: On the weight distribution of two classes of cyclic codes. IEEE Trans. Inf. Theory 54, 5332–5344 (2008).
20.
Luo J., Feng K.: Cyclic codes and sequences from generalized Coulter–Matthews function. IEEE Trans. Inf. Theory 54, 5345–5353 (2008).
21.
Luo J., Tang Y., Wang H.: Cyclic codes and sequences: the generalized Kasami case. IEEE Trans. Inf. Theory 56, 2130–2142 (2010).
22.
Ma C., Zeng L., Liu Y., Feng D., Ding C.: The weight enumerator of a class of cyclic codes. IEEE Trans. Inf. Theory 57, 397–402 (2011).
23.
McEliece R.J.: Irreducible cyclic codes and Gauss sums. In: Combinatorics: Proceedings of the NATO Advanced Study Institute, Breukelen. Part 1: Theory of Designs, Finite Geometry and Coding Theory. Math. Centre Tracts, Math. Centrum, Amsterdam, vol. 55 pp. 179–196 (1974).
24.
McEliece R.J., Rumsey J.H.: Euler products, cyclotomy, and coding. J. Number Theory 4, 302–311 (1972).
25.
Moisio M.: Explicit evaluation of some exponential sums. Finite Fields Appl. 15, 644–651 (2009).
26.
Moisio M., Ranto K., Rintaaho M., Väänänen K.: On the weight distribution of the duals of irreducible cyclic codes, cyclic codes with two zeroes and hyper-Kloosterman codes. Adv. Appl. Discret. Math. 3, 155–164 (2009).
27.
Niho Y.: Multivalued cross-correlation functions between two maximal linear recursive sequence, Ph.D. Dissertation. University Southern California, Los Angeles (1970).
28.
Rao A., Pinnawala N.: A family of two-weight irreducible cyclic codes. IEEE Trans. Inf. Theory 56, 2568–2570 (2010).
29.
Schmidt B., White C.: All two-weight irreducible cyclic codes? Finite Fields Appl. 8, 1–17 (2002).
30.
Schroof R.: Families of curves and weight distribution of codes. Bull. Am. Math. Soc. 32, 171–183 (1995).
31.
Thangaraj A., McLaughlin S.: Quantum codes from cyclic codes over GF(\(4^m\)). IEEE Trans. Inf. Theory 47, 1176–1178 (2001).
32.
van der Vlugt M.: Hasse–Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes. J. Number Theory 55, 145–159 (1995).
33.
Vega G.: Determining the number of one-weight cyclic codes when length and dimension are given. In Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. 4547, pp. 284–293. Springer, Berlin (2007).
34.
Vega G.: The weight distribution of an extended class of reducible cyclic codes. IEEE Trans. Inf. Theory 58, 4862–4869 (2012).
35.
Vega G., Wolfmann J.: New classes of 2-weight cyclic codes. Des. Codes Cryptogr. 42, 327–334 (2007).
36.
Wang B., Tang C., Qi Y., Yang Y., Xu M.: The weight distributions of cyclic codes and elliptic curves. IEEE Trans. Inf. Theory 58, 7253–7259 (2012).
37.
Wolfmann J.: Weight distributions of some binary primitive cyclic codes. IEEE Trans. Inf. Theory 40, 2068–2071 (1994).
38.
Xiong M.: The weight distributions of a class of cyclic codes. Finite Fields Appl. 18, 933–945 (2012).
39.
Xiong M.: The weight distributions of a class of cyclic codes III. Finite Fields Appl. 21, 84–96 (2013).
40.
Xiong M.: The weight distributions of a class of cyclic codes II. Des. Codes Cryptogr. 72, 511–528 (2014).
41.
Yang J., Xiong M., Ding C., Luo J.: Weight distribution of a class of cyclic codes with arbitrary number of zeroes. IEEE Trans. Inf. Theory 59, 5985–5993 (2013).
42.
Yang J., Xia L., Xiong M.: Weight distributions of a class of cyclic codes with arbitrary number of zeroes II. CoRR. arXiv:​1405.​6256 (2014).
43.
Zeng X., Hu L., Jiang W., Yue Q., Cao X.: Weight distribution of a \(p\)-ary cyclic code. Finite Fields Appl. 16, 56–73 (2010).
Metadata
Title
Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes
Authors
Maosheng Xiong
Nian Li
Zhengchun Zhou
Cunsheng Ding
Publication date
01-03-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 3/2016
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-014-0027-5

Other articles of this Issue 3/2016

Designs, Codes and Cryptography 3/2016 Go to the issue

OriginalPaper

A class of quaternary linear codes improving known minimum distances

OriginalPaper

Optimal equi-difference conflict-avoiding codes of weight four

OriginalPaper

Exceptional planar polynomials

OriginalPaper

The weight distribution of a family of -ary cyclic codes

OriginalPaper

Weight enumerator of some irreducible cyclic codes

OriginalPaper

The automorphism group of an extremal [120, 60, 24] code does not contain elements of order 29

Premium Partner

    Image Credits