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New primitive $t$-nomials $(t=3,5)$ over $GF(2)$ whose degree is a Mersenne exponent

Author(s): Toshihiro Kumada; Hannes Leeb; Yoshiharu Kurita; Makoto Matsumoto.
Journal: Math. Comp. 69 (2000), 811-814.
MSC (1991): Primary 11-04, 11T06, 12-04, 12E05
Posted: August 18, 1999
Corrigenda: Math. Comp 71 (2002), 1337-1338
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Abstract: All primitive trinomials over $GF(2)$ with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are $X^{859433}+X^{288477}+1$ and its reciprocal. Also two examples of primitive pentanomials over $GF(2)$ with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

References:

[B]
E. R. Berlekamp, Algebraic coding theory, McGraw-Hill, New York, 1968. MR 38:6873

[H]
J. R. Heringa, H. W. J. Blöte and A. Compagner, New primitive trinomials of Mersenne-exponent degrees for random-number generation, Int. J. Mod. Phys. C vol. 3, No. 3, (1992), 561-564. MR 94a:11113

[HT1]
http://www.mersenne.org/status.htm

[HT2]
http://www.utm.edu:80/research/primes/mersenne.shtml

[K]
Y. Kurita and M. Matsumoto, Primitive t-nomials $(t=3,5)$ over GF(2) whose degree is a Mersenne exponent $\leq 44497$, Math. Comp. vol. 56, No. 194, April (1991), pp. 1-99. MR 91h:11138

[L]
R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge Univ. Press, Cambridge, 1986. MR 88c:11073

[M]
M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation vol. 8, No. 1, January 1998, pp. 3-30.

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Additional Information:

Toshihiro Kumada
Affiliation: Department of Mathematics, Keio University, Yokohama, Japan
Email: kumada@math.keio.ac.jp

Hannes Leeb
Affiliation: Department of Statistics, OR and Computer Methods, University of Vienna, Austria
Email: leeb@smc.univie.ac.at

Yoshiharu Kurita
Affiliation: Hungarian Productivity Center, Budapest, Hungary
Email: ykurit@ibm.net

Makoto Matsumoto
Affiliation: Department of Mathematics, Keio University, Yokohama, Japan
Email: matumoto@math.keio.ac.jp

DOI: 10.1090/S0025-5718-99-01168-0
PII: S 0025-5718(99)01168-0
Keywords: Irreducible polynomials, primitive polynomials, finite field, Mersenne exponent
Received by editor(s): May 19, 1998
Posted: August 18, 1999
Additional Notes: This research was supported by the Austrian Science Foundation (FWF), project no. P11143-MAT
Copyright of article: Copyright 2000, American Mathematical Society

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