Abstract
We construct self-dual codes over small fields \({\mathbb {F}_q}\) with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.
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Arasu K.T., Chen Y.Q., Gulliver T.A., Song W.: Self-dual codes over \({\mathbb {F}_3}\) and negacirculant conference matrices. In: Proceedings 2006 IEEE International Symposium on Information Theory, Seattle, WA, pp. 1301–1304, July 2006.
Assmus E.F. Jr., Mattson H.F. Jr.: On weights in quadratic-residue codes. Discret. Math. 3, 1–20 (1972)
Beenker G.F.M.: A note on extended quadratic residue codes over GF(9) and their ternary images. IEEE Trans. Inf. Theory 30, 403–405 (1984)
Bosma W., Cannon J.J., Playoust C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)
Brouwer A.E.: Bounds on the size of linear codes. In: Pless, V.S., Huffman, W.C.(eds) Handbook of Coding Theory, pp. 295–461. Elsevier, Amsterdam (1998)
Gaborit P.: Quadratic double circulant codes over fields. J. Comb. Theory Ser. A 97, 85–107 (2002)
Gaborit P., Otmani A.: Tables of self-dual codes. Online available at 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)
Gaborit P., Zémor G.: Asymptotic improvement of the Gilbert-Varshamov bound for binary linear codes. IEEE Trans. Inf. Theory 54, 3865–3872 (2000)
Georgiou S., Koukouvinos C.: New self-dual codes over GF(5). In: Walker M. (ed.) Proceedings Cryptography and Coding: 7th IMA International Conference. Lecture Notes in Computer Science, vol. 1746, pp. 63–69. Springer, Heidelberg (1999).
Grassl M.: Bounds on the minimum distance of linear codes. Available online at http://www.codetables.de.
Grassl M.: Searching for linear codes with large minimum distance. In: Bosma, W., Cannon, J.(eds) Discovering Mathematics with Magma—Reducing the Abstract to the Concrete, pp. 287–313. Springer, Heidelberg (2006)
Gulliver T.A.: Optimal double circulant self-dual codes over \({\mathbb {F}_4}\) . IEEE Trans. Inf. Theory 46, 271–274 (2000)
Gulliver T.A., Harada M.: New optimal self-dual codes over GF(7). Graphs Comb. 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.: New nonbinary self-dual codes. IEEE Trans. Inf. Theory 54, 415–417 (2008)
Gulliver T.A., Harada M., Miyabayashi H.: Double circulant and quasi-twisted self-dual codes over \({\mathbb {F}_5}\) and \({\mathbb {F}_7}\) . Adv. Math. Commun. 1, 223–238 (2007)
Gulliver T.A., Harada M., Miyabayashi H.: Double circulant self-dual codes over \({\mathbb {F}_4}\) II. Australas. J. Comb. 39, 163–174 (2007)
Gulliver T.A., Kim J.-L., Lee Y.: New MDS or near-MDS self-dual codes. IEEE Trans. Inf. Theory, 54, 4354–4360 (2008)
Harada M., Holzmann W.H., Kharaghani H., Khorvash M.: Extremal ternary self-dual codes constructed from negacirculant matrices. Graphs Comb. 23, 401–417 (2007)
Harada M., Östergård P.R.J.: Self-dual and maximal self-orthogonal codes over \({\mathbb {F}_7}\) . Discrete Math. 256, 471–477 (2002)
Harada M., Östergård P.R.J.: On the classification of self-dual codes over \({{\mathbb F}_5}\) . Graphs Comb. 19, 203–214 (2003)
Jenson R.A.: A double circulant presentation of quadratic residue codes. IEEE Trans. Inf. Theory 26, 223–227 (1980)
Karlin M.: New binary coding results by circulants. IEEE Trans. Inf. Theory 15, 81–92 (1969)
Kim J.-L., Lee Y.: Euclidean and Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory Ser. A 105, 79–95 (2004)
Lam C.W.H., Pless V.S.: There is no (24,12,10) self-dual quaternary code. IEEE Trans. Inf. Theory 36, 1153–1156 (1990)
Leon J.S.: Computing automorphism groups of error-correcting codes. IEEE Trans. Inf. Theory 28, 496–511 (1982)
Leon J.S., Pless V.S., Sloane N.J.A.: Self-dual codes over GF(5). J. Comb. Theory Ser. A 32, 178–194 (1982)
MacWilliams F.J., Odlyzko A.M., Sloane N.J.A., Ward H.N.: Self-dual codes over GF(4). J. Comb. Theory Ser. A 25, 288–318 (1978)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Majer V.: Algorithmen zur Berechnung von Automorphismen von Codes. Universität Karlsruhe (TH), Diplomarbeit (2006)
Nebe G., Rains E.M., Sloane N.J.A.: Self-Dual Codes and Invariant Theory. Springer, Heidelberg (2006)
Pless V.S.: Symmetry codes over GF(3) and new five-designs. J. Comb. Theory Ser. A 12, 119–142 (1972)
Pless V.S., Tonchev V.D.: Self-dual codes over GF(7). IEEE Trans. Inf. Theory 33, 723–727 (1987)
Rains E.M.: Nonbinary quantum codes. IEEE Trans. Inf. Theory 45, 1827–1832 (1999)
Sloane N.J.A.: Is there a (72,36) d = 16 self-dual code?. IEEE Trans. Inf. Theory 19, 251 (1973)
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Communicated by V.D. Tonchev.
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Grassl, M., Gulliver, T.A. On circulant self-dual codes over small fields. Des. Codes Cryptogr. 52, 57–81 (2009). https://doi.org/10.1007/s10623-009-9267-1
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DOI: https://doi.org/10.1007/s10623-009-9267-1