Abstract
We prove that an [n, k, d] q code \({\mathcal{C}}\) with gcd(d, q) = 1 is extendable if \({\sum_{i \not\equiv 0,d}A_i < (q-1)q^{k-2}}\), where A i denotes the number of codewords of \({\mathcal{C}}\) with weight i. This is a generalization of extension theorems for linear codes by Hill and Lizak (Proceedings of the IEEE International Symposium on Information Theory, Whistler, Canada, 1995) and by Landjev and Rousseva (Probl. Inform. Transm. 42: 319–329, 2006).
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References
Batten L.M.: Combinatorics of Finite Geometries, 2nd edn. Cambridge University Press, Cambridge (1997)
Bierbrauer J.: Introduction to Coding Theory. Chapman and Hall, London (2005)
Hill R.: Optimal linear codes. In: Mitchell, C. (eds) Cryptography and Coding II, pp. 75–104. Oxford University Press, Oxford (1992)
Hill R.: An extension theorem for linear codes. Des. Codes Cryptogr. 17, 151–157 (1999)
Hill R., Kolev E. et al.: A survey of recent results on optimal linear codes. In: Holroyd, F.C. (eds) Combinatorial Designs and their Applications. Research Notes in Mathamatics, vol. 403., pp. 127–152. Chapman & Hall, London (1999)
Hill R., Lizak P.: Extensions of linear codes. In: Proceedings of the IEEE International Symposium on Information Theory, p. 345. Whistler, Canada (1995).
Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn. Clarendon Press, Oxford (1998)
Kanazawa R., Maruta T.: On optimal linear codes over F8. Electronic J. Combin. 18, 34 (2011)
Landjev I., Rousseva A.: An extension theorem for arcs and linear codes. Probl. Inform. Transm. 42, 319–329 (2006)
Maruta T.: On the extendability of linear codes. Finite Fields Appl. 7, 350–354 (2001)
Maruta T.: A new extension theorem for linear codes. Finite Fields Appl. 10, 674–685 (2004)
Maruta T.: Extendability of ternary linear codes. Des. Codes Cryptogr. 35, 175–190 (2005)
Maruta T., Landjev I.N., Rousseva A.: On the minimum size of some minihypers and related linear codes. Des. Codes Cryptogr. 34, 5–15 (2005)
Ward H.N.: Divisibility of codes meeting the Griesmer bound. J. Combin. Theory A 83(1), 79–93 (1998)
Yoshida Y., Maruta T.: An extension theorem for [n, k, d] q codes with gcd(d, q) = 2. Australas. J. Combin. 48, 117–131 (2010)
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Communicated by R. Hill.
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Maruta, T., Yoshida, Y. A generalized extension theorem for linear codes. Des. Codes Cryptogr. 62, 121–130 (2012). https://doi.org/10.1007/s10623-011-9497-x
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DOI: https://doi.org/10.1007/s10623-011-9497-x