A Characterization for the Reducibility of Some Self-reciprocal Binary Pentanomials
نویسنده
چکیده
It is well known that a characterization for the irreducibility of selfreciprocal binary pentanomials does not exist [1, 4]. In this work we divide the self-reciprocal binary pentanomials into four big families, in such a way that all members of one of these families are clearly reducible. Using the Berlekamp Algorithm for the factorization of binary polynomials [2], we prove that all members of a second family are also reducible. More specifically, we present a construction through which it is possible to associate, to each one of the pentanomials in this family, a binary symmetric singular submatrix. As we will see, all the nullities of these submatrices are always odd numbers. This, as we will also see, implies that all pentanomials, in this second family, have an even number of irreducible polynomial factors (and for this reason, all of them are reducible). Mathematics Subject Classification: 11T06
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