The second weight of generalized Reed-Muller codes in most cases

The second weight of generalized Reed-Muller codes

The second weight of the Generalized Reed-Muller code of order d over the finite field with q elements is now known for d < q and d > (n − 1)(q − 1). In this paper, we determine the second weight for the other values of d which are not multiple of q − 1 plus 1. For d = a(q − 1) + 1 we only give an estimate.

On the second weight of generalized Reed-Muller codes

Generalized Reed-Muller Codes

the possible choices for n and k are rather thinly distributed in the class of all pairs (n, k) with k ~ n--and it is, therefore, often inefficient to make use of such codes in concrete situations (that is, when a desired pair (n, k) is far from any achievable pair). We have succeeded in overcoming this difficulty by generalizing the Reed-Muller codes in such a way that they exist for every pai...

On the third weight of generalized Reed-Muller codes

In this paper, we study the third weight of generalized Reed-Muller codes. Using results from [6], we prove under some restrictive condition that the third weight of generalized Reed-Muller codes depends on the third weight of generalized Reed-Muller codes of small order with two variables. In some cases, we are able to determine the third weight and the third weight codewords of generalized Re...

On the Second Weight of Generalized Reed-Muller Codes1

Not much is known about the weight distribution of the generalized Reed-Muller code RMq(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 = ...