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A132450
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First primitive GF(2)[X] polynomials of degree n with at most 5 terms, X^n suppressed.
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4
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1, 3, 3, 3, 5, 3, 3, 29, 17, 9, 5, 83, 27, 43, 3, 45, 9, 39, 39, 9, 5, 3, 33, 27, 9, 71, 39, 9, 5, 83, 9, 197, 83, 281, 5, 387, 83
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OFFSET
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1,2
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COMMENTS
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More precisely: minimum value for X=2 of GF(2)[X] polynomials P[X] with at most 4 terms such that X^n+P[X] is primitive. Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software. The limitation of the number of terms occurs first for a(32), which is 197 representing X^7+X^6+X^2+1, rather than 175 representing X^7+X^5+X^3+X^2+X^1+1. Proof is needed that there exists a primitive GF(2)[X] polynomial P[X] of degree n and at most 5 terms for all positive n.
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LINKS
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EXAMPLE
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a(11)=5, or 101 in binary, representing the GF(2)[X] polynomial X^2+1, because X^11+X^2+1 has no more than 5 terms and X is primitive, contrary to X^11, X^11+1, X^11+X^1, X^11+X^1+1 and X^11+X^2.
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CROSSREFS
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2^n+a(n) belongs to A091250. A132449(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132448 (similar, with no restriction on number of terms). Cf. A132452 (similar, with restriction to exactly 5 terms).
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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