Division of trinomials by pentanomials and orthogonal arrays

Abstract

Consider a maximum-length binary shift-register sequence generated by a primitive polynomial f of degree m. Let C f n denote the set of all subintervals of this sequence with length n, where m < n ≤ 2m, together with the zero vector of length n. Munemasa (Finite fields Appl, 4(3): 252-260, 1998) considered the case in which the polynomial f generating the sequence is a trinomial satisfying certain conditions. He proved that, in this case, C f n corresponds to an orthogonal array of strength 2 that has a property very close to being an orthogonal array of strength 3. Munemasa's result was based on his proof that very few trinomials of degree at most 2m are divisible by the given trinomial f . In this paper, we consider the case in which the sequence is generated by a pentanomial f satisfying certain conditions. Our main result is that no trinomial of degree at most 2m is divisible by the given pentanomial f , provided that Communicated by G. Mullen. The authors are supported by NSERC of Canada. 2 Des. Codes Cryptogr. (2007) 45:1-17f is not in a finite list of exceptions we give. As a corollary, we get that, in this case, C f n corresponds to an orthogonal array of strength 3. This effectively minimizes the skew of the Hamming weight distribution of subsequences in the shift-register sequence.Keywords Shift register sequences · Orthogonal arrays · Polymials over finite fields AMS Classification 94A55 · 05B15 · 11T06

Divisibility of polynomials over finite fields and combinatorial applications

“…Dewar et al [7] suggests extending the results to finite fields other than F 2 . Once we are not in F 2 we can consider binomials (there are no irreducible binomials over F 2 ).…”

“…2, we define notation and give previous results as well as we outline the general methodology. The most important result in this section is a simplification of Munemasa's conditions: we only require irreducible polynomials (or even reducible ones under the condition of no repeated roots) for the minimal polynomial of the LFSR instead of the primitive polynomial condition in previous results [7,21]. We also give some combinatorial applications for our results.…”

“…Previous studies on divisibility of polynomials and combinatorial applications were done for polynomials over the binary field F 2 [7,21]. Let f be a primitive polynomial of degree m over F 2 and let a = (a 0 , a 1 , .…”

“…This paper contains results similar to [7] and [21] but for binomials and trinomials over non-binary fields. In Sect.…”

“…Then, Dewar et al [7] extended this to pentanomials (polynomials with five nonzero terms) dividing trinomials over F 2 to give orthogonal arrays with guaranteed strength 3.…”