On correlation distribution of Niho-type decimation d=3(p^m-1)+1
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The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation d=3(p^m-1)+1 over GF(p^2m) for any prime p ≥ 5. Previously this problem was solved by Xia et al. only for p=2 and p=3 in a series of papers. The main difficulty of this problem for p ≥ 5, as pointed out by Xia et al., is to count the number of codewords of "pure weight" 5 in p-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in p-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When p ≥ 7, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.
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