Binary sequences derived from differences of consecutive quadratic residues
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For a prime p> 5 let q_0,q_1,...,q_(p-3)/2 be the quadratic residues modulo p in increasing order. We study two (p-3)/2-periodic binary sequences (d_n) and (t_n) defined by d_n=q_n+q_n+1 2 and t_n=1 if q_n+1=q_n+1 and t_n=0 otherwise, n=0,1,...,(p-5)/2. For both sequences we find some sufficient conditions for attaining the maximal linear complexity (p-3)/2. Studying the linear complexity of (d_n) was motivated by heuristics of Caragiu et al. However, (d_n) is not balanced and we show that a period of (d_n) contains about 1/3 zeros and 2/3 ones if p is sufficiently large. In contrast, (t_n) is not only essentially balanced but also all longer patterns of length s appear essentially equally often in the vector sequence (t_n,t_n+1,...,t_n+s-1), n=0,1,...,(p-5)/2, for any fixed s and sufficiently large p.
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