Polycyclic codes are a generalization of cyclic and constacyclic codes. Even though they have been known since 1972 received some attention more recently, there not many studies on polycyclic This paper presents an in-depth investigation associated with trinomials. Our results include number facts about trinomials, properties codes, new quantum derived from We also state several conjectures pol...

The present paper is a completion of a previous paper of the same title (Zierler and Brillhart, 1968). In our preceding work 187 of the irreducible trinomials T~.k(x) = x" ~ x k ~1 were left to be tested for primitivity at a later date, even though the requisite complete factorizations of 2 ~ 1 were known (these trinomials were identified in (Zierler and Brillhart, 1968) by a superscript minus ...

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinational design, communication theory and so on. Permutation binomials and trinomials attract people’s interest due to their simple algebraic form and additional extraordinary properties. In t...

In this paper, a generalized Montgomery multiplication algorithm in GF(2m) using the Toeplitz matrix-vector representation is presented. The hardware architectures derived from this algorithm provide low-complexity bit-parallel systolic multipliers with trinomials and pentanomials. The results reveal that our proposed multipliers reduce the space complexity of approximately 15% compared with an...

The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2r − 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give ...

The only primitive trinomials of degree 6972593 over GF(2) are x6972593 + x3037958 + 1 and its reciprocal.

A Montgomery’s algorithm in GF(2) based on the Hankel matrix–vector representation is proposed. The hardware architecture obtained from this algorithm indicates low-complexity bit-parallel systolic multipliers with irreducible trinomials. The results reveal that the proposed multiplier saves approximately 36% of space complexity as compared to an existing systolic Montgomery multiplier for trin...

We give a separation bound for the complex roots of a trinomial f ∈ Z[X ]. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of f ; in particular, it is polynomial in log(deg f). It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of f rat...