Abstract
In this paper, we give some new extremal ternary self-dual codes which are constructed by skew-Hadamard matrices. This has been achieved with the aid of a recently presented modification of a known construction method. In addition, we survey the known results for self-dual codes over GF(5) constructed via combinatorial designs, i.e. Hadamard and skew-Hadamard matrices, and we give a new self-dual code of length 72 and dimension 36 whose minimum weight is 16 over GF(5) for the first time. Furthermore, we give some properties of the generated self-dual codes interpreted in terms of algebraic coding theory, such as the orders of their automorphism groups and the corresponding weight enumerators.
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References
Arasu, K.T., Gulliver, T.A.: Self-dual codes over \(\mathbb{F}_p\) and weighing matrices. IEEE Trans. Inform. Theory 47, 2051–2055 (2001)
Assmus Jr., E.F., Mattson Jr., H.F.: New 5-designs. J. Combin. Theory Ser. A 6, 122–151 (1969)
Bosma, W., Cannon, J.: Handbook of Magma Functions. Version 2.9, Sydney (2002)
Brouwer, A.E.: Bounds on linear codes. In: Pless, V., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 295–461. Elsevier, Amsterdam (1998)
Dougherty, S.T., Gulliver, T.A., Harada, M.: Optimal formally self-dual codes over \(\mathbb{F}_5\) and \(\mathbb{F}_7\). Appl. Algebra Engrg. Comm. Comput. 10, 227–236 (2000)
Gaborit, P.: Quadratic double circulant codes over fields. J. Combin. Theory Ser. A 97, 85–107 (2002)
Gaborit, P.: Tables of self-dual codes, http://www.unilim.fr/pages_perso/philippe.gaborit/SD
Gaborit, P., Otmani, A.: Experimental constructions of self-dual codes. Finite Fields Appl. 9, 372–394 (2003)
Georgiou, S., Koukouvinos, C., Lappas, E.: Self-dual codes over some prime fields constructed from skew-Hadamard matrices. J. Discrete Math. Sci. Cryptogr. 10, 255–266 (2007)
Geramita, A.V., Seberry, J.: Orthogonal Designs. Quadratic Forms and Hadamard Matrices. Lecture Notes in Pure and Applied Mathematics, 45. Marcel Dekker, Inc., New York (1979)
Grassl, M.: Searching for linear codes with large minimum distance. In: Bosma, W., Cannon, J. (eds.) Discovering Mathematics with Magma. Springer, Heidelberg (2006)
Grassl, M.: Bounds on the minimum distance of linear codes, http://www.codetables.de
Gulliver, T.A., Harada, M.: New optimal self-dual codes over GF(7). Graphs Combin. 15, 175–186 (1999)
Gulliver, T.A., Harada, M.: Double circulant self-dual codes over GF(5). Ars Comb. 56, 3–13 (2000)
Gulliver, T.A., Harada, M.: On the minimum weight of codes over \(\mathbb{F}_5\) constructed from certain conference matrices. Des. Codes Cryptogr. 31, 139–145 (2004)
Gulliver, T.A., Harada, M., Miyabayashi, H.: Double circulant self-dual codes over \(\mathbb{F}_5\) and \(\mathbb{F}_7\). Adv. Math. Commun. 1, 223–238 (2007)
Han, S., Kim, J.-L., Lee, H., Lee, Y.: Cubic self-dual codes based on building-up constructions (preprint)
Harada, M.: New extremal ternary self–dual codes. Austral. J. Combin. 17, 133–145 (1998)
Harada, M., Munemasa, A.: There exists no self-dual [24,12,10] code over \(\mathbb{F}_5\). Des. Codes Cryptogr. 52, 125–127 (2009)
Harada, M., Östergård, P.R.J.: On the classification of self-dual codes over \(\mathbb{F}_5\). Graphs Combin 19, 203–214 (2003)
Kim, J.-L., Han, S.: On self-dual codes over \(\mathbb{F}_5\). Des. Codes Cryptogr. 48, 43–58 (2008)
Kim, J.-L., Sole, P.: Skew Hadamard designs and their codes. Des. Codes Cryptogr. 49, 135–145 (2008)
Kotsireas, I.S., Koukouvinos, C.: New skew-Hadamard matrices via computational algebra. Australas. J. Combin. 41, 235–248 (2008)
Kotsireas, I.S., Koukouvinos, C., Simos, D.E.: MDS and near-MDS self-dual codes over large prime fields (submitted for publication)
Koukouvinos, C.: Hadamard matrices of order 4t, t odd positive integer, http://www.math.ntua.gr/~ckoukouv
Koukouvinos, C., Simos, D.E.: Construction of new self-dual codes over GF(5) using skew-Hadamard matrices (to appear in Adv. Math. Commun.)
Koukouvinos, C., Stylianou, S.: On skew-Hadamard matrices. Discrete Math. 308, 2723–2731 (2008)
Leon, J.S., Pless, V., Sloane, N.J.A.: Self-dual codes over GF(5). J. Combin. Theory Ser. A 32, 178–194 (1982)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
MacWilliams, F.J., Mallows, C.L., Sloane, N.J.A.: Generalizations of Gleason’s theorem on weight enumerators of self-dual codes. IEEE Trans. Inform. Theory 18, 794–805 (1972)
Nebe, G., Rains, E.M., Sloane, N.J.A.: Self-dual codes and invariant theory. Springer, Heidelberg (2006)
Pless, V.S., Huffman, W.C., Brualdi, R.A.: An introduction to algebraic codes. In: Pless, V., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 295–461. Elsevier, Amsterdam (1998)
Rains, E.M., Sloane, N.J.A.: Self-dual codes. In: Pless, V., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998)
Sloane, N.J.A.: Self-dual codes and lattices, Relations between combinatorics and other parts of mathematics. In: Proceedings of Symposium on pure mathematics, pp. 273–308. American Mathematics Society, Providence (1979)
Tonchev, V.D.: Codes. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, pp. 517–543. CRC Press, Boca Raton (1996)
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Koukouvinos, C., Simos, D.E. (2009). Self-dual Codes over Small Prime Fields from Combinatorial Designs. In: Bozapalidis, S., Rahonis, G. (eds) Algebraic Informatics. CAI 2009. Lecture Notes in Computer Science, vol 5725. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03564-7_18
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DOI: https://doi.org/10.1007/978-3-642-03564-7_18
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