Swan (Pacific J. Math. 12(3) (1962), 1099-1106) characterized the parity of the number of irreducible factors of trinomials over F 2 . Many researchers have recently obtained Swan-like results on determining the reducibility of polynomials over finite fields. In this paper, we determine the parity of the number of irreducible factors for so-called Type I pentanomial f (x) = x m + x n+1 + x n + x + 1 over F 2 with even n. Our result is based on the Stickelberger-Swan theorem and Newton's formula which is very useful for the computation of the discriminant of a polynomial.
A. Silverberg (IEEE Trans. Inform. Theory 49, 2003) proposed a question on the equivalence of identifiable parent property and traceability property for Reed-Solomon code family. Earlier studies on Silverberg's problem motivate us to think of the stronger version of the question on equivalence of separation and traceability properties. Both, however, still remain open. In this article, we integrate all the previous works on this problem with an algebraic way, and present some new results. It is notable that the concept of subspace subcode of Reed-Solomon code, which was introduced in errorcorrecting code theory, provides an interesting prospect for our topic.
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