On the second weight of generalized Reed-Muller codes
- 125 Downloads
Not much is known about the weight distribution of the generalized Reed-Muller code RM q (s,m) when q > 2, s > 2 and m ≥ 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m – 1)(q – 1) we then find the first s + 1 – (m – 1)(q–1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.
KeywordsFootprint Generalized Reed-Muller code Gröbner basis Hamming weight Weight distribution
AMS classifications11G25 11T71 12.25
Unable to display preview. Download preview PDF.
- 2.Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms, 2nd ed. Springer (1997).Google Scholar
- 5.Geil O., Høholdt T.: On hyperbolic codes. In: Bozta, S., Shparlinski, I. (eds.) Proceedings of the AAECC-14. Lecture Notes in Computer Science 2227, pp. 159–171. Springer (2001).Google Scholar
- 7.MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Eighth Impression, North-Holland (1993)Google Scholar
- 8.McEliece, R.J.: Quadratic forms over finite fields and second-order Reed-Muller codes. JPL Space Programs Summary III, 37–58 (1969)Google Scholar