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Structural properties and enumeration of 1-generator generalized quasi-cyclic codes

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Abstract

Let F q be a finite field of cardinality q, m 1, m 2, . . . , m l be any positive integers, and \({A_i=F_q[x]/(x^{m_i}-1)}\) for i = 1, . . . , l. A generalized quasi-cyclic (GQC) code of block length type (m 1, m 2, . . . , m l ) over F q is defined as an F q [x]-submodule of the F q [x]-module \({A_1\times A_2\times\cdots\times A_l}\). By the Chinese Remainder Theorem for F q [x] and enumeration results of submodules of modules over finite commutative chain rings, we investigate structural properties of GQC codes and enumeration of all 1-generator GQC codes and 1-generator GQC codes with a fixed parity-check polynomial respectively. Furthermore, we give an algorithm to count numbers of 1-generator GQC codes.

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References

  1. Conan J., Séguin G.: Structural properties and enumeration of quasi cyclic codes. Appl. Algebra Engrg. Comm. Commput. 4, 25–39 (1993)

    Article  MATH  Google Scholar 

  2. Dinh H.Q., López-Permouth S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inform. Theory 50, 1728–1744 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Esmaeili M., Yari S.: Generalized quasi-cyclic codes: structrureal properties and code construction. Appl. Algebra Engrg. Comm. Commput. 20, 159–173 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ling S., Solé P.: On the algebra structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inform. Theory 47, 2751–2760 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ling S., Solé P.: On the algebra structure of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30, 113–130 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ling S., Solé P.: On the algebra structure of quasi-cyclic codes III: generator theory. IEEE Trans. Inform. Theory 51, 2692–2700 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Séguin G.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inform. Theory 50, 1745–1753 (2004)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Solomon G., van Tiborg H.C.A.: A connection between block codes and convolutional codes. SIAM J. Appl. Math. 37, 358–369 (2004)

    Article  Google Scholar 

  11. Van V.T., Matsui H., Mita S.: Low complexity encoder for generalized quasi-cyclic codes coming from finite geometries. Communications 2009. ICC’09. IEEE International Conference, pp. 1–6 (2009).

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  1. School of Sciences, Shandong University of Technology, Zibo, Shandong, 255091, People’s Republic of China

    Yonglin Cao

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  1. Yonglin Cao
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Correspondence to Yonglin Cao.

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Communicated by H. van Tilborg.

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Cao, Y. Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60, 67–79 (2011). https://doi.org/10.1007/s10623-010-9417-5

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  • DOI: https://doi.org/10.1007/s10623-010-9417-5

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