Primitive 𝑡-nomials (𝑡=3,5) over 𝐺𝐹(2) whose degree is a Mersenne exponent ≤44497

Kurita, Yoshiharu; Matsumoto, Makoto

doi:10.1090/S0025-5718-1991-1068813-6

Primitive $t$-nomials $(t=3,5)$ over $\textrm {GF}(2)$ whose degree is a Mersenne exponent $\le 44497$
HTML articles powered by AMS MathViewer

by Yoshiharu Kurita and Makoto Matsumoto PDF

Math. Comp. 56 (1991), 817-821 Request permission

Abstract:

All of the primitive trinomials over $GF(2)$ with degree p given by one of the Mersenne exponents 19937, 21701, 23209, and 44497 are presented. Also, one example of a primitive pentanomial over $GF(2)$ is presented for each degree up to 44497 that is a Mersenne exponent. The sieve used is briefly described. A problem is posed which conjectures the number of primitive pentanomials of degree p.

References

Elwyn R. Berlekamp, Algebraic coding theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR 0238597
Rudolf Lidl and Harald Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, 1986. MR 860948
Eugene R. Rodemich and Howard Rumsey Jr., Primitive trinomials of high degree, Math. Comp. 22 (1968), 863–865. MR 238813, DOI 10.1090/S0025-5718-1968-0238813-1
Wayne Stahnke, Primitive binary polynomials, Math. Comp. 27 (1973), 977–980. MR 327722, DOI 10.1090/S0025-5718-1973-0327722-7
E. J. Watson, Primitive polynomials $(\textrm {mod}\ 2)$, Math. Comp. 16 (1962), 368–369. MR 148256, DOI 10.1090/S0025-5718-1962-0148256-1

On primitive trinomials

13

14

Neal Zierler, Primitive trinomials whose degree is a Mersenne exponent, Information and Control 15 (1969), 67–69. MR 244205, DOI 10.1016/S0019-9958(69)90631-7