The structure of linear codes of constant weight
Author:
Jay A. Wood
Journal:
Trans. Amer. Math. Soc. 354 (2002), 10071026
MSC (2000):
Primary 94B05
Published electronically:
October 26, 2001
MathSciNet review:
1867370
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Abstract: In this paper we determine completely the structure of linear codes over of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and prehomogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.
 1.
E.
F. Assmus Jr. and H.
F. Mattson, Errorcorrecting codes: An axiomatic approach,
Information and Control 6 (1963), 315–330. MR 0178997
(31 #3251)
 2.
Kenneth
Bogart, Don
Goldberg, and Jean
Gordon, An elementary proof of the MacWilliams theorem on
equivalence of codes, Information and Control 37
(1978), no. 1, 19–22. MR 0479646
(57 #19067)
 3.
Arrigo
Bonisoli, Every equidistant linear code is a sequence of dual
Hamming codes, Ars Combin. 18 (1984), 181–186.
MR 823843
(87b:94044)
 4.
Alexis
Bonnecaze, Patrick
Solé, and A.
R. Calderbank, Quaternary quadratic residue codes and unimodular
lattices, IEEE Trans. Inform. Theory 41 (1995),
no. 2, 366–377. MR 1326285
(96b:94027), http://dx.doi.org/10.1109/18.370138
 5.
Claude
Carlet, Oneweight 𝑍₄linear codes, Coding
theory, cryptography and related areas (Guanajuato, 1998) Springer,
Berlin, 2000, pp. 57–72. MR 1749448
(2000m:94034)
 6.
I. Constantinescu, W. Heise, and Th. Honold, Monomial extensions of isometries between codes over , Proceedings of the Fifth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT '96) (Sozopol, Bulgaria), Unicorn, Shumen, 1996, pp. 98104.
 7.
A.
Roger Hammons Jr., P.
Vijay Kumar, A.
R. Calderbank, N.
J. A. Sloane, and Patrick
Solé, The 𝑍₄linearity of Kerdock, Preparata,
Goethals, and related codes, IEEE Trans. Inform. Theory
40 (1994), no. 2, 301–319. MR 1294046
(95k:94030), http://dx.doi.org/10.1109/18.312154
 8.
Jessie
MacWilliams, Errorcorrecting codes for multiplelevel
transmission, Bell System Tech. J. 40 (1961),
281–308. MR 0141541
(25 #4945)
 9.
, Combinatorial properties of elementary abelian groups, Ph.D. thesis, Radcliffe College, Cambridge, Mass., 1962.
 10.
Bernard
R. McDonald, Finite rings with identity, Marcel Dekker, Inc.,
New York, 1974. Pure and Applied Mathematics, Vol. 28. MR 0354768
(50 #7245)
 11.
Harold
N. Ward, An introduction to divisible codes, Des. Codes
Cryptogr. 17 (1999), no. 13, 73–79. MR 1714370
(2000j:94039), http://dx.doi.org/10.1023/A:1008302520762
 12.
Harold
N. Ward and Jay
A. Wood, Characters and the equivalence of codes, J. Combin.
Theory Ser. A 73 (1996), no. 2, 348–352. MR 1370137
(96i:94028)
 13.
Jay
A. Wood, Extension theorems for linear codes over finite
rings, Applied algebra, algebraic algorithms and errorcorrecting
codes (Toulouse, 1997) Lecture Notes in Comput. Sci., vol. 1255,
Springer, Berlin, 1997, pp. 329–340. MR 1634126
(99h:94062), http://dx.doi.org/10.1007/3540631631_26
 14.
Jay
A. Wood, Duality for modules over finite rings and applications to
coding theory, Amer. J. Math. 121 (1999), no. 3,
555–575. MR 1738408
(2001d:94033)
 15.
, Linear codes over of constant Euclidean weight, Proceedings of the ThirtySeventh Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, 1999, pp. 895896.
 16.
Jay
A. Wood, Weight functions and the extension theorem for linear
codes over finite rings, Finite fields: theory, applications, and
algorithms (Waterloo, ON, 1997), Contemp. Math., vol. 225, Amer.
Math. Soc., Providence, RI, 1999, pp. 231–243. MR 1650644
(2000b:94024), http://dx.doi.org/10.1090/conm/225/03225
 17.
, Understanding linear codes of constant weight using virtual linear codes, Proceedings of the ThirtyEighth Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, 2000, pp. 10381046.
 18.
, The structure of linear codes of constant weight, Proceedings of the International Workshop on Coding and Cryptography, Paris, INRIA, 2001, pp. 547556.
 1.
 E. F. Assmus, Jr. and H. F. Mattson, Errorcorrecting codes: An axiomatic approach, Inform. and Control 6 (1963), 315330. MR 31:3251
 2.
 K. Bogart, D. Goldberg, and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes, Inform. and Control 37 (1978), 1922. MR 57:19067
 3.
 A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984), 181186. MR 87b:94044
 4.
 A. Bonnecaze, P. Solé, and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41 (1995), 366377. MR 96b:94027
 5.
 C. Carlet, Oneweight linear codes, Coding Theory, Cryptography and Related Areas (J. Buchmann, T. Høholdt, H. Stichtenoth, and H. TapiaRecillas, eds.), Springer, Berlin, 2000, pp. 5772. MR 2000m:94034
 6.
 I. Constantinescu, W. Heise, and Th. Honold, Monomial extensions of isometries between codes over , Proceedings of the Fifth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT '96) (Sozopol, Bulgaria), Unicorn, Shumen, 1996, pp. 98104.
 7.
 A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), 301319. MR 95k:94030
 8.
 F. J. MacWilliams, Errorcorrecting codes for multiplelevel transmission, Bell System Tech. J. 40 (1961), 281308. MR 25:4945
 9.
 , Combinatorial properties of elementary abelian groups, Ph.D. thesis, Radcliffe College, Cambridge, Mass., 1962.
 10.
 B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, vol. 28, Marcel Dekker, Inc., New York, 1974. MR 50:7245
 11.
 H. N. Ward, An introduction to divisible codes, Des. Codes Cryptogr. 17 (1999), 7379. MR 2000j:94039
 12.
 H. N. Ward and J. A. Wood, Characters and the equivalence of codes, J. Combin. Theory Ser. A 73 (1996), 348352. MR 96i:94028
 13.
 J. A. Wood, Extension theorems for linear codes over finite rings, Applied Algebra, Algorithms and ErrorCorrecting Codes (T. Mora and H. Mattson, eds.), Lecture Notes in Comput. Sci., vol. 1255, SpringerVerlag, Berlin, 1997, pp. 329340. MR 99h:94062
 14.
 , Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), 555575. MR 2001d:94033
 15.
 , Linear codes over of constant Euclidean weight, Proceedings of the ThirtySeventh Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, 1999, pp. 895896.
 16.
 , Weight functions and the extension theorem for linear codes over finite rings, Finite fields: Theory, Applications and Algorithms (R. C. Mullin and G. L. Mullen, eds.), Contemp. Math., vol. 225, Amer. Math. Soc., Providence, 1999, pp. 231243. MR 2000b:94024
 17.
 , Understanding linear codes of constant weight using virtual linear codes, Proceedings of the ThirtyEighth Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, 2000, pp. 10381046.
 18.
 , The structure of linear codes of constant weight, Proceedings of the International Workshop on Coding and Cryptography, Paris, INRIA, 2001, pp. 547556.
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Additional Information
Jay A. Wood
Affiliation:
Department of Mathematics, Computer Science & Statistics, Purdue University Calumet, Hammond, Indiana 46323, Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, and GRIM, Université ToulonVar, 83957 La Garde Cedex, France
Address at time of publication:
Department of Mathematics, Western Michigan University, 1903 W. Michigan Ave., Kalamazoo, Michigan 49008–5248
Email:
jay.wood@wmich.edu
DOI:
http://dx.doi.org/10.1090/S0002994701029051
PII:
S 00029947(01)029051
Keywords:
Constant weight codes,
Lee weight,
Euclidean weight,
extension theorem,
orbital codes,
virtual codes
Received by editor(s):
January 15, 2001
Published electronically:
October 26, 2001
Additional Notes:
Partially supported by Purdue University Calumet Scholarly Research Awards. Some results were announced in [17] and [18]. Theorem 10.3 first appeared in [15]
Article copyright:
© Copyright 2001
American Mathematical Society
