We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The considered are seen to be determined up change bases a single number. Studying this number leads new family polynomials which exhibit special properties. These satisfy three term recursion and closely related zigzag zero-one sequences. Interpreting the for "prime" 1 yields classical...

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some fam...

For each integer [Formula: see text], we identify new infinite families of monogenic trinomials text] with non-squarefree discriminant, many which have small Galois group. Moreover, in certain situations when fixed and produce asymptotics on the number such text].

(a) f has two distinct roots with identical norm v if and only if p is located on a real double point of the hypotrochoid, (b) f has a root of multiplicity 2 with norm v if and only if the corresponding hypotrochoid is a hypocycloid and p is a cusp of it, and (c) f has more than two roots with norm v if and only if p = 0 if and only if the hypotrochoid is a rhodonea curve with a multiple point ...