## On the period length of pseudorandom vector sequences generated by matrix generators.(English)Zbl 0657.65008

Let m be a positive integer, let $${\mathbb{Z}}_ m=\{0,1,...,m-1\}$$, and let A be an $$r\times r$$ matrix with elements in $${\mathbb{Z}}_ m$$ which is nonsingular (mod m). The following linear recursive congruential matrix generator for generating r-dimensional pseudorandom vectors is considered $$(*)\quad \bar x_{n+1}\equiv A\bar x_ n (mod m)$$, $$\bar x_{n+1}\in {\mathbb{Z}}^ r_ m$$, $$n\geq 0$$, with $$\bar x_ 0\in {\mathbb{Z}}^ r_ m$$. The sequence $$\{z_ n:$$ $$n\geq 0\}$$ is purely periodic and let $$\lambda(A,\bar x_ 0,m)$$ denote its period length. In this paper the case $$m=p^{\alpha}$$, $$\alpha\geq 2$$, is considered with p a prime number. It is shown that for $$p\geq 3$$ and $$r\geq 2$$ there exist matrix generators (*) with $$\lambda (A,\bar x_ 0,p^{\alpha})=(p^ r- 1)p_{\alpha -1}$$ for any $$\bar x_ 0\in {\mathbb{Z}}^ r_{p^{\alpha}}$$ with $$\bar x_ 0\equiv \bar O (mod p)$$.
Reviewer: R.Theodorescu

### MSC:

 65C10 Random number generation in numerical analysis 11K99 Probabilistic theory: distribution modulo $$1$$; metric theory of algorithms
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### References:

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