The Covering Radius of the Reed--Muller Code RM(2,7) is 40
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It was proved by J. Schatz that the covering radius of the second order Reed--Muller code RM(2, 6) is 18 (IEEE Trans Inf Theory 27: 529--530, 1985). However, the covering radius of RM(2,7) has been an open problem for many years. In this paper, we prove that the covering radius of RM(2,7) is 40, which is the same as the covering radius of RM(2,7) in RM(3,7). As a corollary, we also find new upper bounds for RM(2,n), n=8,9,10.
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