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A Note on Constacyclic and Quasi-cyclic Codes over \({\mathbb {F}}_2+u{\mathbb {F}}_2+v{\mathbb {F}}_2+uv{\mathbb {F}}_2\)

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Abstract

In this paper, we study \((1+\lambda u)\)-constacyclic codes of length \(2^k\) over the ring \(\mathrm {R}={\mathbb {F}}_2+u{\mathbb {F}}_2+v{\mathbb {F}}_2+uv{\mathbb {F}}_2\), where \(u^2=v^2=0,\ uv=vu\), k is a positive integer and \(\lambda \) is a unit in \(\mathrm {R}\). We classify all \((1+\lambda u)\)-constacyclic codes of length \(2^k\) over \(\mathrm {R}\). We also completely classify the structure of \((1+\lambda u)\)-constacyclic codes of length \(2^k\) over \(\mathrm {R}\) that are contained in their annihilators as well as equal to their annihilators. We enumerate these codes and present mass formulas for them. Some optimal codes are obtained as the Gray images of these codes. In addition, we also study the structure of 1-generator generalized quasi-cyclic codes over the ring \(\mathrm {R}\). A minimal spanning set for these codes is determined. A BCH-type bound on the minimum distance of these codes is also presented.

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References

  1. Abualrub, T., Siap, T.: Constacyclic codes over \(\mathbb{F}_2 + u\mathbb{F}_2\). J. Frankl. Inst. 346(5), 520–529 (2009)

    Article  MATH  Google Scholar 

  2. Aydin, N., Gulliver, T.A.: Some good cyclic and quasi-twisted \(\mathbb{Z}_4\)-linear codes. Arts Comb. 99, 503–518 (2011)

    MATH  Google Scholar 

  3. Aydin, N., Karadeniz, S., Yildiz, B.: Some new binary quasi-cyclic codes from codes over the ring \(\mathbb{F}_2 + u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2\). Appl. Algebra Eng. Commun. Comput. 24(5), 355–367 (2013)

    Article  Google Scholar 

  4. 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)

    Article  MATH  Google Scholar 

  5. Bhaintwal, M., Wasan, S.K.: On quasi-cyclic codes over \(\mathbb{Z}_q\). Appl. Algebra Eng. Commun. Comput. 20, 459–480 (2009)

    Article  MATH  Google Scholar 

  6. Cao, Y.: Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration. Appl. Algebra Eng. Commun. Comput. 22, 219–233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castillo-Guillén, C., Rentería-Márquez, C., Tapia-Recillas, H.: Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index \(3\). Finite Fields Appl. 43, 1–21 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castillo-Guillén, C., Rentería-Márquez, C., Tapia-Recillas, H.: Duals of constacyclic codes over finite local Frobenius non-chain rings of length \(4\). Discrete Math. 341(4), 919–933 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dinh, H.Q.: Constacyclic codes of length \(2^s\) over Galois extension of rings \(\mathbb{F}_2 + u\mathbb{F}_2\). IEEE Trans. Inf. Theory 55(4), 1730–1740 (2009)

    Article  Google Scholar 

  10. Dinh, H.Q.: Constacyclic codes of length \(p^s\) over \(\mathbb{F}_{p^m} + u\mathbb{F}_{p^m}\). J. Algebra 324(5), 940–950 (2010)

    Article  MathSciNet  Google Scholar 

  11. Dougherty, S.T., Kaya, A., Saltürk, E.: Cyclic codes over local Frobenius rings of order \(16\). Adv. Math. Commun. 11(1), 99–114 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Esmaeili, M., Gulliver, T.A., Secord, N.P., Mahmoud, S.A.: A link between quasi-cyclic codes and convolutional codes. IEEE Trans. Inf. Theory 44(1), 431–435 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Esmaeili, M., Yari, S.: Generalized quasi-cyclic codes: structural properties and codes construction. Appl. Algebra Eng. Commun. Comput. 20(2), 159–173 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gao, J., Kong, Q.: \(1\)-Generator quasi-cyclic codes over \(\mathbb{F}_{p^m} +u\mathbb{F}_{p^m} +\cdots +u^{s-1}\mathbb{F}_{p^m}\). J. Frankl. Inst. 350(10), 3260–3276 (2013)

    Article  MATH  Google Scholar 

  15. Grassl, M., Table of Bounds on Linear Codes (Online). http://www.codetables.de (1995). Accessed on 10 May 2018

  16. Haifeng, Y., Zhu, S., Kai, X.: \((1-uv)\)-Constacyclic codes over \({\mathbb{F}}_p + u{\mathbb{F}}_p + v{\mathbb{F}}_p + uv{\mathbb{F}}_p\). J. Syst. Sci. Complex. 27(4), 811–816 (2014)

    Article  MathSciNet  Google Scholar 

  17. Karadeniz, S., Yildiz, B.: \((1 + v)\)-Constacyclic codes over \({\mathbb{F}}_2 + u{\mathbb{F}}_2 + v{\mathbb{F}} _2 + uv{\mathbb{F}}_2\). J. Frankl. Inst. 348(9), 2625–2632 (2011)

    Article  Google Scholar 

  18. Kai, X., Zhu, S., Wang, L.: A family of constacyclic codes over \({\mathbb{F}}_2 + u{\mathbb{F}}_2 + v{\mathbb{F}}_2 + uv{\mathbb{F}}_2\). J. Syst. Sci. Complex. 25(5), 1032–1040 (2012)

    Article  MathSciNet  Google Scholar 

  19. Kai, X.S., Zhu, S.X., Li, P.: \((1+\lambda u)\)-Constacyclic codes over \({\mathbb{F}}_p[u]/{u^m}\). J. Frankl. Inst. 347(5), 751–762 (2010)

    Article  MATH  Google Scholar 

  20. Kasami, T.: A Gilbert–Varshamov bound for quasi-cyclic codes of rate \(1/2\). IEEE Trans. Inf. Theory 20(5), 679–679 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, Y., Shi, M., Solé, P.: Two-weight and three-weight codes from trace codes over \({\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{{F}}_p\). Discrete Math. 341(2), 350–357 (2018)

    Article  MathSciNet  Google Scholar 

  22. Liu, X., Xu, X.: Some class of repeated-root constacyclic codes over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+u^2{\mathbb{F}}_{p^m}\). J. Korean Math. Soc. 51(4), 853–866 (2014)

    Article  MathSciNet  Google Scholar 

  23. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. Elsevier, New York (1977)

    MATH  Google Scholar 

  24. Martínez-Moro, E., Szabo, S.: On codes over local Frobenius non-chain rings of order \(16\). Contemp. Math. 634, 227–241 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Norton, G.H., Sălăgean, A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebra Eng. Commun. Comput. 10(6), 489–506 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Qian, J.F., Zhang, L.N., Zhu, S.X.: \((1 + u)\)-Constacyclic and cyclic codes over \({\mathbb{F}}_2 + u{\mathbb{F}}_2\). Appl. Math. Lett. 19(8), 820–823 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Siap, I., Abualrub, T., Aydin, N.: Quaternary quasi-cyclic codes with even length components. ARS Comb. 101, 425–434 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Siap, I., Kulhan, N.: The structure of generalized quasi cyclic codes. Appl. Math. E Notes. 5, 24–30 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Siap, I., Abualrub, T., Yildiz, B.: One generator quasi-cyclic codes over \({\mathbb{F}}_2+ u{\mathbb{F}}_2\). J. Frankl. Inst. 349(1), 284–292 (2012)

    Article  MATH  Google Scholar 

  30. Shi, M., Guan, Y., Solé, P.: Two new families of two-weight codes. IEEE Trans. Inf. Theory 63(10), 6240–6246 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shi, M., Qian, L., Solé, P.: Few-weight codes from trace codes over a local ring. Appl. Algebra Eng. Commun. Comput. 29(4), 335–350 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    B Srinivasulu & Maheshanand Bhaintwal

  2. Department of Mathematics, Dr. SPM International Institute of Information Technology, Naya Raipur, India

    Rama Krishna Bandi

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  1. B Srinivasulu
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  2. Rama Krishna Bandi
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  3. Maheshanand Bhaintwal
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Correspondence to B Srinivasulu.

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Communicated by Miin Huey Ang.

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Srinivasulu, B., Bandi, R.K. & Bhaintwal, M. A Note on Constacyclic and Quasi-cyclic Codes over \({\mathbb {F}}_2+u{\mathbb {F}}_2+v{\mathbb {F}}_2+uv{\mathbb {F}}_2\). Bull. Malays. Math. Sci. Soc. 42, 3453–3474 (2019). https://doi.org/10.1007/s40840-019-00745-5

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  • DOI: https://doi.org/10.1007/s40840-019-00745-5

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