Abstract
Quaternary 1-generator quasi-cyclic codes are considered in the paper. Under the conditions that n is odd and gcd(|2| n , m) = 1, where |2| n denotes the order of 2 modulo n, we give the enumeration of quaternary 1-generator quasi-cyclic codes of length mn, and describe an algorithm which will obtain one, and only one, generator for each quaternary 1-generator quasi-cyclic code.
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Bhargava V.K., Séguin G.E., Stein J.M.: Some (mk, k) cyclic codes in quasi-cyclic form. IEEE Trans. Inform. Theory 24, 630–632 (1978)
Dey B.K., Rajan B.S.: IF q -linear cyclic codes over \({IF_{q^m}}\): DFT characterization. In: Bozatas S., Shparlinski I.E. (eds.) Lecture Notes in Computer Science, vol. 2227, 67–76 (2001).
Gulliver T.A., Bhargava V.K.: Two new rate 2/P binary quasi-cyclic codes. IEEE Trans. Inform. Theory 40, 1667–1668 (1994)
Hammons A.R., Kumar P.V. Jr., Calderbank A.R., Sloane N.J.A., Solé P.: The ℤ 4-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)
Lally N.J.A.K., Fitzpatrick P.: Algebraic structure of quasi-cyclic codes. Discrete Appl. Math. 246, 157–175 (2001)
Ling S., Solé P.: On the algebraic structure quasi-cyclic codes I: Finite fields. IEEE Trans. Inform. Theory 47, 2751–2760 (2001)
Ling S., Solé P.: On the algebraic structure quasi-cyclic codes II: chain rings. Des. Codes Crypotogr. 30, 113–130 (2003)
Menezes A.J.: Applications of Finite Fields. Klumer Academic Publishers, Boston (1993)
Nechaev A.A.: Kerdock code in a cyclic form. Diskretnaya Mat. (USSR) 1, 123–139 (1989)
Séguin G.E.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inform. Theory 50, 1745–1753 (2004)
Tanner R.M.: A transformation theory for a class of group invariant codes. IEEE Trans. Inform. Theory 34, 752–775 (1998)
Tavares S.E., Bhargava V.K., Shiva S.G.S.: Some rate P/P + 1 quasi-cyclic codes. IEEE Trans. Inform. Theory 20, 133–135 (1974)
Wan Z.-X.: Cyclic codes over Galois rings. Algebra Colloquium 6(3), 291–304 (1999)
Wan Z.-X.: Quaternary Codes. World Scientific, Singapore (1997)
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Communicated by R. Hill.
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Cui, J., Pei, J. Quaternary 1-generator quasi-cyclic codes. Des. Codes Cryptogr. 58, 23–33 (2011). https://doi.org/10.1007/s10623-010-9381-0
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DOI: https://doi.org/10.1007/s10623-010-9381-0