# American Institute of Mathematical Sciences

February  2016, 10(1): 45-52. doi: 10.3934/amc.2016.10.45

## On extendability of additive code isometries

 1 IMATH, Université de Toulon, B.P. 20132, 83957 La Garde, France

Received  November 2014 Revised  July 2015 Published  March 2016

For linear codes, the MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a linear isometry of the whole space. But, in general, this is not the situation for nonlinear codes. In this paper codes over a vector space alphabet are considered. It is proved that if the length of such code is less than some threshold value, then an analogue of the MacWilliams Extension Theorem holds. One family of unextendable code isometries for the threshold value of code length is described.
Citation: Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
##### References:
 [1] S. V. Avgustinovich and F. I. Solov'eva, To the metrical rigidity of binary codes,, Probl. Inf. Transm., 39 (2003), 178. doi: 10.1023/A:1025148221096. Google Scholar [2] K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes,, Inf. Control, 37 (1978), 19. doi: 10.1016/S0019-9958(78)90389-3. Google Scholar [3] R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes,, J. Combin. Theory, 1 (1966), 96. doi: 10.1016/S0021-9800(66)80007-8. Google Scholar [4] I. Constantinescu and W. Heise, On the concept of code-isomorphy,, J. Geometry, 57 (1996), 63. doi: 10.1007/BF01229251. Google Scholar [5] H. Q. Dinh and S. R. López-Permouth, On the equivalence of codes over rings and modules,, Finite Fields Appl., 10 (2004), 615. doi: 10.1016/j.ffa.2004.01.001. Google Scholar [6] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247. doi: 10.1142/S0219498804000873. Google Scholar [7] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory Ser. A, 92 (2000), 17. doi: 10.1006/jcta.1999.3033. Google Scholar [8] J. Gruska, Quantum Computing,, McGraw-Hill, (1999). Google Scholar [9] D. I. Kovalevskaya, On metric rigidity for some classes of codes,, Probl. Inf. Transm., 47 (2011), 15. doi: 10.1134/S0032946011010029. Google Scholar [10] J. Luh, On the representation of vector spaces as a finite union of subspaces,, Acta Math. Acad. Sci. Hungar., 23 (1972), 341. doi: 10.1007/BF01896954. Google Scholar [11] F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups,, Ph.D thesis, (1962). Google Scholar [12] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes,, North-Holland, (1977). Google Scholar [13] F. Solov'eva, T. Honold, S. Avgustinovich and W. Heise, On the extendability of code isometries,, J. Geometry, 61 (1998), 2. doi: 10.1007/BF01237489. Google Scholar [14] H. N. Ward and J. A. Wood, Characters and the equivalence of codes,, J. Combin. Theory Ser. A, 73 (1996), 348. doi: 10.1016/S0097-3165(96)80011-2. Google Scholar [15] J. A. Wood, Duality for modules over finite rings and applications to coding theory,, Amer. J. Math., 121 (1999), 555. doi: 10.1353/ajm.1999.0024. Google Scholar [16] J. A. Wood, Foundations of linear codes defined over finite modules: The extension theorem and the MacWilliams identities,, in Codes over Rings (ed. P. Sóle), (2009), 124. doi: 10.1142/9789812837691_0004. Google Scholar

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##### References:
 [1] S. V. Avgustinovich and F. I. Solov'eva, To the metrical rigidity of binary codes,, Probl. Inf. Transm., 39 (2003), 178. doi: 10.1023/A:1025148221096. Google Scholar [2] K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes,, Inf. Control, 37 (1978), 19. doi: 10.1016/S0019-9958(78)90389-3. Google Scholar [3] R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes,, J. Combin. Theory, 1 (1966), 96. doi: 10.1016/S0021-9800(66)80007-8. Google Scholar [4] I. Constantinescu and W. Heise, On the concept of code-isomorphy,, J. Geometry, 57 (1996), 63. doi: 10.1007/BF01229251. Google Scholar [5] H. Q. Dinh and S. R. López-Permouth, On the equivalence of codes over rings and modules,, Finite Fields Appl., 10 (2004), 615. doi: 10.1016/j.ffa.2004.01.001. Google Scholar [6] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247. doi: 10.1142/S0219498804000873. Google Scholar [7] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory Ser. A, 92 (2000), 17. doi: 10.1006/jcta.1999.3033. Google Scholar [8] J. Gruska, Quantum Computing,, McGraw-Hill, (1999). Google Scholar [9] D. I. Kovalevskaya, On metric rigidity for some classes of codes,, Probl. Inf. Transm., 47 (2011), 15. doi: 10.1134/S0032946011010029. Google Scholar [10] J. Luh, On the representation of vector spaces as a finite union of subspaces,, Acta Math. Acad. Sci. Hungar., 23 (1972), 341. doi: 10.1007/BF01896954. Google Scholar [11] F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups,, Ph.D thesis, (1962). Google Scholar [12] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes,, North-Holland, (1977). Google Scholar [13] F. Solov'eva, T. Honold, S. Avgustinovich and W. Heise, On the extendability of code isometries,, J. Geometry, 61 (1998), 2. doi: 10.1007/BF01237489. Google Scholar [14] H. N. Ward and J. A. Wood, Characters and the equivalence of codes,, J. Combin. Theory Ser. A, 73 (1996), 348. doi: 10.1016/S0097-3165(96)80011-2. Google Scholar [15] J. A. Wood, Duality for modules over finite rings and applications to coding theory,, Amer. J. Math., 121 (1999), 555. doi: 10.1353/ajm.1999.0024. Google Scholar [16] J. A. Wood, Foundations of linear codes defined over finite modules: The extension theorem and the MacWilliams identities,, in Codes over Rings (ed. P. Sóle), (2009), 124. doi: 10.1142/9789812837691_0004. Google Scholar
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