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

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We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant of elliptic curves. Our proof does not use the circle method. We use Laplace's method of steepest descent and the Hardy–Ramanujan asymptotic formula for the partition function. (The latter asymptotic formula can be derived without the circle method.)

Abstract. We re-prove the Hardy-Ramanujan asymptotic formula for the partition function without using the circle method. We derive our result from recent work of Bruinier and Ono on harmonic weak Maass forms.

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