Some new binary quasi-cyclic codes from codes over the ring \(\mathbb F _2+u\mathbb F _2+v\mathbb F _2+uv\mathbb F _2\)

  • Nuh Aydin
  • Suat Karadeniz
  • Bahattin YildizEmail author
Original Paper


We consider quasi-cyclic codes over the ring \(\mathbb{F }_2+u\mathbb{F }_2+v\mathbb{F }_2+uv\mathbb{F }_2\), a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions.


Lee weight Gray maps Cyclic codes Quasi-cyclic codes 

Mathematics Subject Classification (2000)

94B15 94B05 13P05 



The authors would like to thank the anonymous referees for their valuable remarks and suggestions that improved the presentation of this paper.


  1. 1.
    Aydin, N., Gulliver, T.A.: Some good cyclic and quasi-twisted \({\mathbb{Z}}_4\)-linear codes. Ars Comb. 99, 503–518 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aydin, N., Ray-Chaudhuri, D.K.: Quasi-cyclic codes over \({\mathbb{Z}}_4\) and some new binary codes. IEEE Trans. Inf. Theory 48(7), 2065–2069 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aydin, N., Siap, I., Ray-Chaudhuri, D.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 23(3), 313–326 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bini, G., Flamini, F.: Finite Commutative Rings and Applications. Kluwer Academic Publishers, Boston (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Z.: A Database of Binary Quasi-cyclic Codes. Online available at Accessed 28 July 2011
  7. 7.
    Daskalov, R., Hristov, P.: New binary one-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 49(11), 3001–3005 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dougherty, S.T., Yildiz, B., Karadeniz, S.: Codes over \(R_k\). Gray maps and their binary images. Finite Fields Appl. 17(3), 205–219 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grassl, M.: Bound on the Minimum Distance of Linear Codes and Quantum Codes. Online available at Accessed on 28 July 2011.
  10. 10.
    Gulliver, T.A., Bhargava, V.K.: Nine good rate \((m-1)/pm\) quasi-cyclic codes. IEEE Trans. Inf. Theory 38(4), 1366–1369 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gulliver, T.A., Bhargava, V.K.: Some best rate \( 1/p\) and rate \((p-1)/p\) systematic quasi-cyclic codes over GF(3) and GF(4). IEEE Trans. Inf. Theory 38(4), 1369–1374 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Heijnen, P., von Tilborg, H., Verhoeff, T., Weijs, S.: Some new binary quasi-cyclic codes. IEEE Trans. Inf. Theory 44(5), 1994–1996 (1998)CrossRefzbMATHGoogle Scholar
  13. 13.
    Karadeniz, S., Yildiz, B.: Double-circulant and double-bordered-circulant constructions for self-dual codes over \(R_2\). Adv. Math. Commun. 6(2), 193–202 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Siap, I., Abualrub, T., Aydin, N.: Quaternary quasi-cyclic codes with even length components. Ars Comb. 101, 425–434 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yildiz, B., Karadeniz, S.: Linear codes over \({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}\). Des. Codes Cryptogr. 54(1), 61–81 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yildiz, B., Karadeniz, S.: Cyclic Codes over \({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}\). Des. Codes Cryptogr. 58(3), 221–234 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsKenyon CollegeGambierUSA
  2. 2.Department of MathematicsFatih UniversityIstanbulTurkey

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