THE MACWILLIAMS IDENTITY FOR LIPSCHITZ WEIGHT ENUMERATORS

Abstract

In this paper, the MacWilliams identity is stated for codes
over quaternion integers with respect to the Lipschitz metric.

Keywords

• MacWilliams identity,Block codes,Weight enumerator,Lipschitz metric.

References

1. F. J. MacWilliams, ”Combinatorial Problems of Elementary Abelian Groups,” Ph.D.
2. dissertation, Harvard Univ., Cambridge, MA, 1962.
3. F. J. MacWilliams, ”A theorem on the distribution of weights in a systematic code,”
4. Bell Syst. Tech. J., vol. 42, pp. 79-94, 1963.
5. F. J. Macwilliams and N. J. Sloane, ”The Theory of Error Correcting Codes”, North
6. Holland Pub. Co., 1977.
7. S. Irfan, ”MacWilliams identity for m− spotty Lee weight enumerators,” Appl. Math.
8. Lett., 23 (2010) 13-16.

9. B. Yildiz and S. Karadeniz., ”Linear codes over F2 + uF2 + vF2 + uvF2,” Des. Codes
10. Cryptogr., (2010) 54:6181.
11. J. A. Wood, Duality for modules over finite rings and applications to coding theory,
12. Amer. J. Math. 121 (1999), 555575.
13. T. Honold and I. Landjev, MacWilliams Identities for codes over frobenius ring, Finite
14. Fields and application Springer pp. 276-292, 2000.
15. V. Zinoviev, T. Ericson, On Fourier-Invariant Partitions of Finite Abelian Groups and
16. the MacWilliams Identity for Group Codes, Problems of Information Transmission, 32,
17. No. 1, 1996.
18. K. Huber, ”The MacWilliams Theorem for Two-Dimensional Modulo Metrics,”
19. AAECC, 41-48, 1997.
20. C. Martinez, E. Stafford, R. Beivide and E. Gabidulin. ”Perfect Codes over Lipschitz
21. Integers”. IEEE Int. Symposium on Information Theory, ISIT’07.
22. C. Martinez, R. Beivide, and E. M. Gabidulin, ”Perfect Codes from Cayley Graphs over
23. Lipschitz Integers,” IEEE Trans. Inform.Theory, vol. 55, pp. 3552-3562, August, 2009.
24. G. Davidoff, P. Sarnak, A. Valette, ”Elementary Number Theory, Group Theory, and
25. Ramanujan Graphs”, Cambridge University Pres, 2003.
26. M. G¨uzeltepe, ”Codes over Hurwitz integers”, Vol. 313/5, pp. 704-714, 2013.
27. M. G¨uzeltepe, O. Heden, ”Perfect Mannheim, Lipschitz and Hurwitz weight codes”,
28. Math. Communications, Vol. 19/2 pp. 253-276, 2014.
29. O. Heden, M. G¨uzeltepe, ”On perfect 1-E error-correcting codes”, Math. Communications, Vol. 20/1 pp. 23-35, 2015.