Abstract
By applying a method for constructing binary self-dual codes with an automorphism of odd prime order \(p\), we give a full classification of all optimal binary self-dual codes of length 50 having an automorphism of order 3. As a consequence, we give a full classification of all \([50, 25, 10]\) codes possessing an automorphism of odd prime order. Up to equivalence, there are exactly 177,601 such codes. This completely determines all possibilities for the cardinality of the automorphism group of such a code. Also, we show that there are at least 52 non-isomorphic quasi-symmetric 2-(49, 9, 6) designs, derived from the \([50,25,10]\) codes with a particular weight enumerator.
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References
Bouyukliev I.: About the Code Equivalence, Advances in Coding Theory and Cryptography. Series on Coding Theory and Cryptology, vol. 3. World Scientific Publishing, Singapore (2007).
Bouyuklieva S., Bouyukliev I.: An algorithm for classification of binary self-dual codes. IEEE Trans. Inf. Theory 58(6), 3933–3940 (2012).
Bouyuklieva S., Harada M.: Extremal self-dual codes with automorphisms of order 3 and quasi-symmetric 2-(49, 9, 6) designs. Des. Codes Cryptogr. 28(2), 163–169 (2003).
Bouyuklieva S., Yankov N., Kim J.L.: Classification of binary self-dual [48, 24, 10] codes with an automorphism of odd prime order. Finite Fields Appl. 18(6), 1104–1113 (2012).
Bouyuklieva S., Yankov N., Russeva R.: Classification of the binary self-dual [42, 21, 8] codes having an automorphism of order 3. Finite Fields Appl. 13, 605–615 (2007).
Bouyuklieva S., Yankov N., Russeva R.: On the classification of binary self-dual [44, 22, 8] codes with an automorphism of order 3 or 7. Int. J. Inf. Coding Theory 2(1), 21–37 (2011).
Conway J.H., Pless V., Sloane N.J.A.: Self-dual codes over GF(3) and GF(4) of length not exceeding 16. IEEE Trans. Inf. Theory 25(3), 312–322 (1979).
Harada M., Gulliver T., Kaneta H.: Classification of extremal double-circulant self-dual codes of length up to 62. Discret. Math. 188(1–3), 127–136 (1998).
Harada M., Munemasa A.: Some restrictions on weight enumerators of singly even self-dual codes. IEEE Trans. Inf. Theory 52(3), 1266–1269 (2006).
Huffman W.C.: Automorphisms of codes with applications to extremal doubly even codes of length 48. IEEE Trans. Inf. Theory 28(3), 511–521 (1982).
Huffman W.C.: On the classification and enumeration of self-dual codes. Finite Fields Appl. 11(3), 451–490 (2005).
Huffman W.C., Tonchev V.D.: The existence of extremal self-dual [50, 25, 10] codes and quasi-symmetric 2-(49, 9, 6) designs. Des. Codes Cryptogr. 6(2), 97–106 (1995).
Kim H.J.: The binary extremal self-dual codes of lengths 38 and 40. Des. Codes Cryptogr. 63(1), 43–57 (2012).
MacWilliams F., Odlyzko A., Sloane N., Ward H.: Self-dual codes over GF(4). J. Comb. Theory Ser. A 25(3), 288–318 (1978).
Pless V.: A classification of self-orthogonal codes over GF(2). Discret. Math. 3, 209–246 (1972).
Pless V., Sloane N.: On the classification and enumeration of self-dual codes. J. Comb. Theory Ser. A 18(3), 313–335 (1975).
Rains E., Sloane N.: Self-dual codes. In: Pless V.S., Huffman W.C., editors. Handbook of Coding Theory. Amsterdam: Elsevier; (1998).
Yankov N., Lee M.H.: New binary self-dual codes of lengths 50–60. Des. Codes Cryptogr. (2013).
Yorgov V.: A method for constructing inequivalent self-dual codes with applications to length 56. IEEE Trans. Inf. Theory 33(1), 77–82 (1987).
Acknowledgments
We thank the anonymous referees for the comments and suggestions, which contributed to improving the quality of the publication. This paper was written with the support of the Ministry of Education Science and Technology (MEST) and the Korean Federation of Science and Technology Societies (KOFST). This work was also supported by World Class University Project (WCU) R-32-2012-000-20014-0, Basic Science Research Program 2010-0020942, NRF Korea, and MEST 2012-002521, NRF Korea. The first author was also supported by Shumen University under Grant RD-08-245/13.03.2013.
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Communicated by J.-L. Kim.
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Yankov, N., Lee, M.H. Classification of self-dual codes of length 50 with an automorphism of odd prime order. Des. Codes Cryptogr. 74, 571–579 (2015). https://doi.org/10.1007/s10623-013-9874-8
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DOI: https://doi.org/10.1007/s10623-013-9874-8