

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
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OFFSET

1,5


COMMENTS

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 maximumlength linear feedback shift registers with efficient implementation in both hardware and software.


LINKS

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)


EXAMPLE

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.


CROSSREFS

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


KEYWORD

more,sign


AUTHOR

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


STATUS

approved



