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On the structure of the linear codes with a given automorphism

The research is partially supported by [the Bulgarian National Science Fund under Contract No KP-06-H62/2/13.12.2022]

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  • The purpose of this paper is to present the structure of the linear codes over a finite field with $ q $ elements that have a permutation automorphism of order $ m $. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual.

    Mathematics Subject Classification: Primary: 94B05, 20B25.

    Citation:

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  • [1] R. AnsteeM. Hall Jr and J. G. Thompson, Planes of order 10 do not have a collineation of order 5, Journ. Combin. Theory ser. A, 29 (1980), 39-58.  doi: 10.1016/0097-3165(80)90046-1.
    [2] E. F. Assmus and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151. 
    [3] E. R. Berlekamp, Coding theory and the Mathieu groups, Info. Control, 18 (1971), 40-64. 
    [4] S. D. Berman, Semisimple cyclic and Abelian codes. Ⅱ, Kibernetika (Kiev), 3 (1967), 21-30 (Russian). doi: 10.1007/BF01119999.
    [5] M. Borello, On automorphism groups of binary linear codes, in Topics in Finite Fields, Contemporary Mathematics, 632 (2015), 29-42.
    [6] M. Borello and G. Nebe, On involutions in extremal self-dual codes and the dual distance of semi self-dual codes, Finite Fields and Their Applications, 33 (2015), 80-89.  doi: 10.1016/j.ffa.2014.11.008.
    [7] M. Borello and W. Willems, Automorphisms of order $2p$ in binary self-Dual extremal codes of length a multiple of 24, IEEE Trans. Inform. Theory, 59 (2013), 3378-3383. 
    [8] I. BouyuklievV. Fack and J. Winne, $2-(31, 15, 7)$, $2-(35, 17, 8)$ and $2-(36, 15, 6)$ designs with automorphisms of odd prime order, and their related Hadamard matrices and codes, Designs, Codes and Cryptography, 51 (2009), 105-22.  doi: 10.1007/s10623-008-9247-x.
    [9] S. Buyuklieva, On the binary self-dual codes with an automorphism of order 2, Designs, Codes and Cryptography, 12 (1997), 39-48. 
    [10] S. Buyuklieva, A method for constructing self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 46 (2000), 496-504.  doi: 10.1109/18.825812.
    [11] S. Buyuklieva and J. de la Cruz, On the structure of binary LCD codes having an automorphism of odd prime order, IEEE Trans. Inform. Theory, 68 (2022), 6426-6433. 
    [12] S. BuyuklievaA. Malevich and W. Willems, Automorphisms of extremal self-dual codes, IEEE Trans. Inform. Theory, 56 (2010), 2091-2096.  doi: 10.1109/TIT.2010.2043763.
    [13] J. H. Conway and V. Pless, On primes dividing the group order of a doubly-even (72, 36, 16) code and the group order of a quaternary (24, 12, 10) code, Discrete Mathematics, 38 (1982), 143-156.  doi: 10.1016/0012-365X(82)90284-9.
    [14] S. T. DoughertyP. GaboritM. Harada and P. Sole, Type Ⅱ codes over $ \mathbb{F}_2+u \mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45. 
    [15] M. Esmaeili and S. Yari, Generalized quasi-cyclic codes: Structural properties and code construction, Appl. Algebra Eng. Commun. Comput., 20 (2009), 159-173. 
    [16] M. Esmaeili and S. Yari, On complementary-dual quasi-cyclic codes, Finite Fields and Their Applications, 15 (2009), 375-386.  doi: 10.1016/j.ffa.2009.01.002.
    [17] C. GüneriF. ÖzbudakB. ÖzkayaE. SaçikaraZ. Sepasdar and P. Solé, Structure and performance of generalized quasi-cyclic codes, Finite Fields and Their Applications, 47 (2017), 183-202.  doi: 10.1016/j.ffa.2017.06.005.
    [18] S. H. HoughtonC. W. H. LamL. H. Thiel and J. A. Parker, The extended quadratic residue code is the only $[48, 24, 12]$ self-dual doubly-even code, IEEE Trans. Inform. Theory, 49 (2003), 53-59. 
    [19] W. C. Huffman, Decomposing and shortening codes using automorphisms, IEEE Trans. Inform. Theory, 32 (1986), 833-836.  doi: 10.1109/TIT.1986.1057233.
    [20] W. C. Huffman, Automorphisms of codes with applications to extremal doubly even codes of length 48, IEEE Trans. Inform. Theory, 28 (1982), 511-521. 
    [21] W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28, Ⅰ, IEEE Trans. Inform. Theory, 36 (1990), 651-660. 
    [22] W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28, Ⅱ, IEEE Trans. Inform. Theory, 37 (1991), 1206-1216.  doi: 10.1109/18.86976.
    [23] W. C. Huffman, On extremal self-dual ternary codes of lengths 28 to 40, IEEE Trans. Inform. Theory, 38 (1992), 1395-1400.  doi: 10.1109/18.144724.
    [24] W. C. Huffman, Self-dual $ \mathbb{F}_q$-linear $ \mathbb{F}_{q^t}$-codes with an automorphism of prime order, Advances in Mathematics of Communications, 7 (2013), 57-90. 
    [25] W. Knapp and P. Schmid, Codes with prescribed permutation group, J. Algebra, 67 (1980), 415-435. 
    [26] S. LingH. Niederreiter and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅳ: Repeated roots, Designs, Codes and Cryptography, 38 (2006), 337-361.  doi: 10.1007/s10623-005-1431-7.
    [27] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅰ: Finite fields, IEEE Trans. Inform. Theory, 47 (2001), 2751-2760.  doi: 10.1109/18.959257.
    [28] F. J. MacWilliams, Codes and ideals in group algebras, Combinatorial Mathematics and its Applications, 317 (1969), 317-328. 
    [29] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, North-Holland, Amsterdam, 1977.
    [30] G. Nebe, On extremal self-dual ternary codes of length 48, International Journal of Combinatorics, 2012 (2012), 9 pp. doi: 10.1155/2012/154281.
    [31] V. Pless, On the uniqueness of the Golay codes, J. Combin. Theory, 5 (1968), 215-228. 
    [32] V. Pless and W. C. Huffman, Handbook of Coding Theory, Elsevier, Amsterdam, 1998.
    [33] I. Siap and N. Kulhan, The structure of generalized quasi-cyclic codes, Appl. Math. E-Notes, 5 (2005), 24-30. 
    [34] N. J. A. Sloane, Is there a (72, 36), $d=16$ self-dual code?, IEEE Trans. Inform. Theory, 19 (1973), 251.  doi: 10.1109/tit.1973.1054975.
    [35] V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inform. Theory, 33) (1987), 77-82.  doi: 10.1109/TIT.1987.1057273.
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