Two remarks on a paper by Moreno and Kumar [binary Reed-Muller codes]

O. Moreno and P.V. Kumar (IEEE Trans. Inform. Theory, vol.39, no.5, September 1993) showed how Deligne´s theorem can be applied to coding theory. They studied certain subcodes of binary Reed-Muller codes and estimated the associated character sums over a field of q² elements. They obtained bounds of the order 𝒪(q). The present paper shows that in one case one can improve the coefficient of q in the estimates. The author also shows that there is an error in Moreno and Kumar´s argument and in some cases one needs need to replace a bound of the order 𝒪(q) by a weaker bound of the order 𝒪(q ^3/2)