On triple-sum-sets and two or three weights codes (Q795016)

The authors introduce and characterize triple-sum-sets which are a natural generalization of partial difference sets (which in turn are connected to binary uniformly packed codes and two-weight codes [\textit{J. Wolfmann}, J. Comb. Theory, Ser. A 23, 208-222 (1977; Zbl 0364.94020); \textit{P. Camion}, Difference sets in elementary Abelian groups (1979; Zbl 0455.94018)]). They also obtain a characterization of the perfect 3-error- correcting Golay code and of strongly uniformly packed linear 2-error- correcting binary codes in terms of triple-sum-sets. The transposed codes of a given projective code are also defined and the possible weights of these codes are given with some applications. Finally the authors give a construction of a triple-sum-set in the binary case. They use this construction to prove that there exists a triple-sum-set whose characteristic function is a coset leader of the first order Reed-Muller code R(1,k) of length \(2^ k\) and whose cardinality attains the lower bound given by \textit{T. Helleseth}, \textit{T. Kløve} and \textit{J. Mykkeltveit} [IEEE Trans. Inf. Theory IT-24, 627-628 (1978; Zbl 0379.94017)] for the covering radius of R(1,k).