Abstract
For linear codes, the MacWilliams Extension Theorem states that each linear isometry of a code extends to a linear isometry of the whole space. But, in general, it is not the situation for nonlinear codes. In the paper it was proved, that if the length of an additive code is less than some threshold value, then an analogue of the MacWilliams Extension Theorem holds. One family of unextendible code isometries for the threshold value of code length is described.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Avgustinovich, S.V., Solov’eva, F.I.: To the metrical rigidity of binary codes. Probl. Inf. Trans. 39(2), 178–183 (2003). doi:10.1023/A:1025148221096
Clark, P.L.: Covering numbers in linear algebra. Am. Math. Mon. 119(1), 65–67 (2012). doi:10.4169/amer.math.monthly.119.01.065
Constantinescu, I., Heise, W.: On the concept of code-isomorphy. J. Geom. 57(1–2), 63–69 (1996). doi:10.1007/BF01229251
Gruska, J.: Quantum Computing. McGraw-Hill, London (1999)
Kovalevskaya, D.I.: On metric rigidity for some classes of codes. Probl. Inf. Trans. 47(1), 15–27 (2011). doi:10.1134/S0032946011010029
MacWilliams, F., Sloane, N.: The Theory of Error-Correcting Codes: Vol.: 1. North-Holland Mathematical Library. North-Holland, Amsterdam/London/New York (1977)
Solov’eva, F., Honold, T., Avgustinovich, S., Heise, W.: On the extendability of code isometries. J. Geom. 61(1–2), 2–16 (1998). doi:10.1007/BF01237489
Ward, H.N., Wood, J.A.: Characters and the equivalence of codes. J. Comb. Theory Ser. A 73(2), 348–352 (1996). doi:10.1016/S0097-3165(96)80011-2
Wood, J.A.: Exotic automorphisms of additive codes. AMS Sectional Meeting, Louisville (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Dyshko, S. (2015). On Extendibility of Additive Code Isometries. In: Pinto, R., Rocha Malonek, P., Vettori, P. (eds) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-17296-5_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-17296-5_17
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17295-8
Online ISBN: 978-3-319-17296-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)