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

show all references

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

[1]

Heide Gluesing-Luerssen. On isometries for convolutional codes. Advances in Mathematics of Communications, 2009, 3 (2) : 179-203. doi: 10.3934/amc.2009.3.179

[2]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543

[3]

Manish K. Gupta, Chinnappillai Durairajan. On the covering radius of some modular codes. Advances in Mathematics of Communications, 2014, 8 (2) : 129-137. doi: 10.3934/amc.2014.8.129

[4]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028

[5]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025

[6]

Otávio J. N. T. N. dos Santos, Emerson L. Monte Carmelo. A connection between sumsets and covering codes of a module. Advances in Mathematics of Communications, 2018, 12 (3) : 595-605. doi: 10.3934/amc.2018035

[7]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399

[8]

Karim Samei, Saadoun Mahmoudi. Singleton bounds for R-additive codes. Advances in Mathematics of Communications, 2018, 12 (1) : 107-114. doi: 10.3934/amc.2018006

[9]

David Keyes. $\mathbb F_p$-codes, theta functions and the Hamming weight MacWilliams identity. Advances in Mathematics of Communications, 2012, 6 (4) : 401-418. doi: 10.3934/amc.2012.6.401

[10]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

[11]

Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251

[12]

Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399

[13]

Evangeline P. Bautista, Philippe Gaborit, Jon-Lark Kim, Judy L. Walker. s-extremal additive $\mathbb F_4$ codes. Advances in Mathematics of Communications, 2007, 1 (1) : 111-130. doi: 10.3934/amc.2007.1.111

[14]

Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425

[15]

W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2008, 2 (3) : 309-343. doi: 10.3934/amc.2008.2.309

[16]

W. Cary Huffman. Additive cyclic codes over $\mathbb F_4$. Advances in Mathematics of Communications, 2007, 1 (4) : 427-459. doi: 10.3934/amc.2007.1.427

[17]

Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042

[18]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83

[19]

Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329

[20]

W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]