swartz@watdcsu.waterloo.edu ( SWARTZ SJ - COMBINATORICS Swartz OPT. ) (09/14/89)
C & O SEMINAR
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|DATE: Friday, September 15, 1989 |
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|TIME: 3:30 p.m. |
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|PLACE: MC4041 |
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|SPEAKER: Dr. R. Wilson |
| Department of Mathematics, Caltech|
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|TITLE: The minimum distance of some |
| polynomial codes |
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ABSTRACT
Of fundamental interest in coding theory is the
question of what can be said about the weight (number
of nonzero coefficients) of a polynomial f(x) of
degree less than n, given that f(x) has certain n-th
roots of unity among its zeros. We will review some
joint work with J.H. van Lint that produces lower
bounds on the minimum weight and which generalizes the
BCH, the Hartmann-Tseng, and the Roos bounds. We then
specialize to polynomials over Gf(2) and lengths
r
n=2 -1, and ask when the binary codes of polynomials
t
whose zeros include < and < (where < is a primitive
r
element of GF(2 )) have minimum distance 5. When t=3,
we have the classical two-error-correcting BCH codes.
In general, this question is open. However, our
bounds provide other examples, e.g., t=5 and t=13
whenever r is odd.