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
Cryptography and Communications

09-05-2024 | Research

Additive codes with few weights

Authors: Daniel Panario, Murat Sahin, Qiang Wang

Published in: Cryptography and Communications

Log in

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

search-config
loading …

Abstract

Additive codes have a wide range of applications. A classical nice and generic way to construct linear codes is via trace functions. In this paper, first, we generalize this method to construct additive codes. Then, we use this method to get some explicit additive codes. Computing Weil-like sums, we obtain parameters of these codes such as the length and weight distribution. We show that our codes have few weights.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
2.
Bachoc, C., Gaborit, P.: On extremal additive GF(4)-codes of lengths 10 to 18. J. Theorie Nombres Bordeaux 12, 225–271 (2000)
3.
Bierbrauer, J., Bartoli, D., Faina, G., Marcugini, S., Pambianco, F.: The nonexistence of an additive quaternary [15,5,9]-code. Finite Fields Their Appl. 36, 29–40 (2015)MathSciNetCrossRef
4.
Blaum, M., Brady, J., Bruck, J., Menon, J.: EVENODD: An efficient scheme for tolerating double disk failures in RAID architectures. IEEE Trans. Comput. 44, 192–202 (1995)CrossRef
5.
6.
Blaum, M., Bruck, J., Vardy, A.: MDS array codes with independent parity symbols. IEEE Trans. Inf. Theory 42, 529–542 (1995)CrossRef
7.
Blaum, M., Farrell, P.G., van Tilborg, H.C.A.: Array codes in Handbook of Coding Theory. North-Holland, Amsterdam 1(2), 1855–1909 (1998)
8.
Blaum, M., Roth, R.M.: New array codes for multiple phased burst correction. IEEE Trans. Inf. Theory 39, 66–77 (1993)MathSciNetCrossRef
9.
10.
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)MathSciNetCrossRef
11.
Cardell, S.D., Climent, J.J., Requena, V.: A construction of MDS array codes. WIT Trans. Inf. Commun. Technol. 45, 47–58 (2013)
12.
Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15, 125–156 (1998)MathSciNetCrossRef
13.
Carlet, C., Ding, C., Yuan, J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51, 2089–2102 (2005)MathSciNetCrossRef
14.
Coulter, R.S.: On the evaluation of a class of Weil sums in characteristics 2. New Zealand J. Math. 28, 171–184 (1999)MathSciNet
15.
Danielsen, L.E., Parker, M.G.: Directed graph representation of half-rate additive codes over GF(4). Des. Codes Cryptogr. 59, 119–130 (2011)MathSciNetCrossRef
16.
17.
18.
Ding, K., Ding, C.: Binary linear codes with three weights. IEEE Commun. Lett. 18, 1879–1882 (2014)CrossRef
19.
Ding, K., Ding, C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61, 5835–5842 (2015)MathSciNetCrossRef
20.
21.
Ding, C., Yuan, J.: Covering and secret sharing with linear codes. In: International Conference on Discrete Mathematics and Theoretical Computer Science (DMTCS); Dijon, France. 11–25, (2003)
22.
Gaborit, P., Huffman, W. C., Kim, J. L., Pless, V.: On the classification of extremal additive codes over GF(4). In Proceedings of the 37th Allerton Conference on Communication, Control, and Computing, Urbana-Champaign, 535–544, (1999)
23.
Gaborit, P., Huffman, W. C., Kim, J. L., Pless, V.: On additive GF(4) codes. DIMACS Workshop on Codes and Association Schemes, DIMACS Series in Discrete Math. and Theoret. Computer Science, Amer. Math. Soc., Providence, RI, vol. 56, 135–149, (2001)
24.
Gibson, G.A.: Redundant Disk Arrays: Reliable, Parallel Secondary Storage. MIT Press, Cambridge, MA (1992)
25.
Heng, Z., Yue, Q., Li, C.: Three classes of linear codes with two or three weights. Discrete Math. 339, 2832–2847 (2016)MathSciNetCrossRef
26.
27.
Jian, G., Lin, Z., Feng, R.: Two-weight and three-weight linear codes based on Weil sums. Finite Fields Appl. 57, 92–107 (2019)MathSciNetCrossRef
28.
Kim, J.L., Lee, N.: Secret sharing schemes based on additive codes over GF(4). Appl. Algebra Engrg. Comm. Comput. 28, 79–97 (2017)MathSciNetCrossRef
29.
Li, C., Yue, Q., Fu, F.: A construction of several classes of two-weight and three-weight linear codes. Appl. Algebra Engrg. Comm. Comput. 28, 11–30 (2017)MathSciNetCrossRef
30.
31.
Lidl, R., Mullen, G.L., Turnwald, G.: Dickson Polynomials. Pitman Monographs Surv. Pure Appl. Math. 65, (1993)
32.
Louidor, E., Roth, R.M.: Lowest density MDS codes over extension alphabets. IEEE Trans. Inf. Theory 52, 3186–3197 (2006)MathSciNetCrossRef
33.
Li, N., Mesnager, S.: Recent results and problems on constructions of linear codes from cryptographic functions. Cryptogr. Commun. 12, 965–986 (2020)MathSciNetCrossRef
34.
Mesnager, S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptogr. Commun. 9, 71–84 (2017)MathSciNetCrossRef
35.
Mesnager, S., Özbudak, F., Sinak, A.: Linear codes from weakly regular plateaued functions and their secret sharing schemes. Des. Codes Cryptogr. 87, 463–480 (2019)MathSciNetCrossRef
36.
Tang, C., Li, N., Qi, Y., Zhou, Z., Helleseth, T.: Linear codes with two or three weights from weakly regular bent functions. IEEE Trans. Inf. Theory 62, 1166–1176 (2016)MathSciNetCrossRef
37.
Tang, D., Carlet, C., Zhou, Z.: Binary linear codes from vectorial boolean functions and their weight distribution. Discrete Math. 340, 3055–3072 (2017)MathSciNetCrossRef
38.
Wolfmann, J.: Polynomial description of binary linear codes and related properties. Appl. Algebra Engrg. Comm. Comput. 2, 119–138 (1991)MathSciNetCrossRef
39.
Wu, Y., Li, N., Zeng, X.: Linear codes with few weights from cyclotomic classes and weakly regular bent functions. Des. Codes Cryptogr. 88, 1255–1272 (2020)MathSciNetCrossRef
40.
41.
Xiang, C., Feng, K., Tang, C.: A Construction of linear codes over \(\mathbb{F} _2\) from Boolean functions. IEEE Trans. Inf. Theory 63, 169–176 (2017)CrossRef
42.
43.
Yuan, J., Carlet, C., Ding, C.: The weight distributions of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52, 712–716 (2006)MathSciNetCrossRef
44.
Zaitzev, G.V., Zinovev, V.A., Semakov, N.V.: Minimum-check-density codes for correcting bytes of errors, erasures, or defects. Probl. Inf. Transm. 19, 197–204 (1983)MathSciNet
Metadata
Title
Additive codes with few weights
Authors
Daniel Panario
Murat Sahin
Qiang Wang
Publication date
09-05-2024
Publisher
Springer US
Published in
Cryptography and Communications
Print ISSN: 1936-2447
Electronic ISSN: 1936-2455
DOI
https://doi.org/10.1007/s12095-024-00720-3

Premium Partner

    Image Credits