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A132454 First primitive GF(2)[X] polynomials of degree n and minimal number of terms, expressed as -k for X^n+X^k+1, else with X^n suppressed. 3
1, -1, -1, -1, -2, -1, -1, 29, -4, -3, -2, 83, 27, 43, -1, 45, -3, -7, 39, -3, -2, -1, -5, 27, -3, 71, 39, -3, -2, 83, -3, 197, -13, 281, -2, -11, 83 (list; graph; refs; listen; history; text; internal format)



More precisely: when there exists k, 0<k<n, such that X^n+X^k+1 is a GF(2)[X] primitive polynomial, negative of the minimum of such k; else minimum value for X=2 of GF(2)[X] polynomials P[X] such that X^n+P[X] is primitive and has the minimum number of terms for a primitive polynomials of degree n. The special encoding of trinomials greatly extends the range of a(n) representable using a given number of bits; for example a(89) = -38 instead of 2^38+1. Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software.


Table of n, a(n) for n=1..37.

Index entries for sequences operating on GF(2)[X]-polynomials

Index entries for sequences related to trinomials over GF(2)


a(10)=-3, representing the GF(2)[X] polynomial X^10+X^3+1, because this degree 10 trinomial is primitive, contrary to X^10+X^1+1, X^10+X^2+1 and X^10+X^2+X^1.


Either of 2^n+2^(-a(n))+1 or 2^n+a(n) belongs to A091250. If there exists m such that n = A073726(m), then a(n) = -A074744(m); otherwise a(n) = A132450(n). A132453(n) gives the primitive polynomial corresponding to a(n). Cf. A132448, similar with no restriction on number of terms. Cf. A132450, similar with restriction to at most 5 terms. Cf. A132452, similar with restriction to exactly 5 terms.

Sequence in context: A246072 A147802 A093076 * A182911 A058293 A172092

Adjacent sequences:  A132451 A132452 A132453 * A132455 A132456 A132457




Francois R. Grieu (f(AT)grieu.com), Aug 22 2007



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Last modified August 25 06:01 EDT 2019. Contains 326323 sequences. (Running on oeis4.)