OPTIMAL LINEAR CODES OVER ℤm

Title & Authors
OPTIMAL LINEAR CODES OVER ℤm
Dougherty, Steven T.; Gulliver, T. Aaron; Park, Young-Ho; Wong, John N.C.;

Abstract
We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring $\small{{\mathbb{Z}}_m}$. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over $\small{{\mathbb{Z}}_8}$ and $\small{{\mathbb{Z}}_9}$ of lengths up to 6. We determine the minimum distances of optimal linear codes over $\small{{\mathbb{Z}}_4}$ for lengths up to 7. Some examples of optimal codes are given.
Keywords
linear codes;optimal codes;codes over rings;
Language
English
Cited by
1.
MDS and self-dual codes over rings, Finite Fields and Their Applications, 2012, 18, 6, 1061
2.
The number of self-dual codes over $${Z_{p^3}}$$, Designs, Codes and Cryptography, 2009, 50, 3, 291
3.
MDS codes over finite principal ideal rings, Designs, Codes and Cryptography, 2009, 50, 1, 77
4.
The classification of self-dual modular codes, Finite Fields and Their Applications, 2011, 17, 5, 442
References
1.
T. Abualrub and R. Oehmke, On the generators of $Z_4$ cyclic codes of length $2^e$, IEEE Trans. Inform. Theory 49 (2003), no. 9, 2126-2133

2.
J. M. P. Balmaceda, A. L. Rowena, and F. R. Nemenzo, Mass formula for self-dual codes over $Z_{p2}$, Discrete Math. (to appear)

3.
T. Blackford, Cyclic codes over $Z_4$ of oddly even length, Discrete Appl. Math. 128 (2003), no. 1, 27-46

4.
A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1995), no. 1, 21-35

5.
J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo 4, J. Combin, Theory Ser. A 62 (1993), no. 1, 30-45

6.
S. T. Dougherty, T. A. Gulliver, and J. N. C. Wong, Self-dual codes over$Z_8\;and\;Z_9$, Des. Codes Cryptogr. 41 (2006), no. 3, 235-249

7.
S. T. Dougherty, M. Harada, and P. Sole, Self-dual codes over rings and the Chinese remainder theorem, Hokkaido Math. J. 28 (1999), no. 2, 253-283

8.
S. T. Dougherty and S. Ling, Cyclic codes over $Z_4$ of even length, Des. Codes Cryptogr. 39 (2006), no. 2, 127-153

9.
S. T. Dougherty, S. Y. Kim, and Y. H. Park, Lifted codes and their weight enumerators, Discrete Math. 305 (2005), no. 1-3, 123-135

10.
S. T. Dougherty and Y. H. Park, Codes over the p-adic integers, Des. Codes Cryptogr. 39 (2006), no. 1, 65-80

11.
S. T. Dougherty and Y. H. Park, On modular cyclic codes, Finite Fields Appl. 13 (2007), no. 1, 31-57

12.
S. T. Dougherty and K. Shiromoto, MDR codes over $Z_k$, IEEE Trans. Inform. Theory 46 (2000), no. 1, 265-269

13.
S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Inform. Theory 47 (2001), no. 1, 400-404

14.
S. T. Dougherty and T. A. Szczepanski, Latin k-hypercubes, submitted

15.
J. Fields, P. Gaborit, J. S. Leon, and V. Pless, All self-dual $Z_4$ codes of length 15 or less are known, IEEE Trans. Inform. Theory 44 (1998), no. 1, 311-322

16.
M. Greferath, G. McGuire, and M. O'Sullivan, On Plotkin-optimal codes over finite F'robenius rings, J. Algebra Appl. 5 (2006), no. 6, 799-815

17.
T. A. Gulliver and J. N. C. Wong, Classification of Optimal Linear $Z_4$ Rate 1/2 Codes of Length $\leq$ 8, submitted

18.
R. Hill, A First Course in Coding Theory, Oxford Applied Mathematics and Computing Science Series. The Clarendon Press, Oxford University Press, New York, 1986

19.
Y. H. Park, Modular Independence and Generator Matrices for Codes over $Z_m$, submitted

20.
W. C. Huffman and V. S. Pless, Fundamentals of Error-correcting Codes, Fundamentals of error-correcting codes. Cambridge University Press, Cambridge, 2003

21.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, NorthHolland, Amsterdam, 1977

22.
G. H. Norton and A. Salagean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory 46 (2000), no. 3, 1060-1067

23.
V. Pless, J. S. Leon, and J. Fields, All $Z_4$ codes of type II and length 16 are known, J. Combin. Theory Ser. A 78 (1997), no. 1,32-50

24.
K. Shiromoto and L. Storrne, A Griesmer Bound for Linear Codes over Finite QuasiFrobenius Rings, Discrete Appl, Math. 128 (2003), no. 1,263-274

25.
J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555-575