The weight distribution of C-5(1,n)

In [2] the codes C-q (r, n) over F-q were introduced. These linear codes ha ve parameters [2(n), Sigma (r)(i=0) ((n)(i)), 2(n-r)], can be viewed as ana logues of the binary Reed-Muller codes and share several features in common with them. In [2], the weight distribution Of C-3 (1, n) is completely det ermined. In this paper we compute the weight distribution of C-5(1, n). To this end we transform a sum of a product of two binomial coefficients into an expres sion involving only exponentials and Lucas numbers. We prove an effective r esult on the set of Lucas numbers which are powers of two to arrive to the complete determination of the weight distribution of C-5 (1, n). The final result is stated as Theorem 2.