Abstract. We derive pseudorandom binary sequences from maximal length sequences over the integral residue rings. We prove that these derived binary sequences have guaranteed large periods, and we also obtain upper bounds on their minimal polynomials in the sense of the partial order defined by divisibility.

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On the linear complexity A(Z) of a periodically repeated random bit sequence 2, R. Rueppel proved that, for two extreme cases of the period T, the expected linear complexity E[A(Z)] is almost equal to T, and suggested that E[A(Z)] would be close to T in general [6, pp. 33-521 [7, 81. In this note we obtain bounds of E[A(Z)], as well as bounds of the variance Var[A(Z)], both for the general case of T, and we estimate the probability distribution of A(Z). Our results on E[A(Z)] quantify the closeness of E[A(Z)] and T, in particular, formally confirm R. Rueppel's suggestion.

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1. Introduction. Continued fractions [8,14] are a useful tool in many number theoretical problems and in numerical computing. It is well known that the simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. Many people have contrived to construct multidimensional continued fractions in dealing with the rational approximation problem for multi-reals. One construction is the Jacobi-Perron algorithm (JPA) (see [1]). This algorithm and its modifications have been extensively studied [6,7,10,13]. These algorithms are adapted to study the same problem for multi-formal Laurent series [2,4,11,12]. But none of them guarantees the best rational approximation in general. In this paper, we deal with the multi-rational approximation problem over the formal Laurent series field F ((z −1 )): given an element r ∈ F ((z −1 )) m , find p ∈ F [z] m and q ∈ F [z] such that p/q approximates r as close as possible while deg(q) is bounded.We propose a new continued fraction algorithm for multi-formal Laurent series. It is proved that this algorithm guarantees best rational approximations for multi-formal Laurent series.The paper is organized as follows: Section 2 deals with the indexed valuation of F ((z −1 )) m . Section 3 contains the detailed definition of the problem of optimal rational approximation of multi-formal Laurent series. Section 4 proposes an algorithm called multidimensional continued fraction algorithm (m-CFA, for short), which produces a multi-continued fraction expansion C(r) for any given multi-formal Laurent series r. Section 5 shows that C(r) satisfies three basic conditions. Section 6 states the main results of this pa-