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                          C & O  SEMINAR
          =============================================
          |                                            |
          |DATE:     Friday, September 15, 1989        |
          |                                            |
          |TIME:     3:30 p.m.                         |
          |                                            |
          |PLACE:    MC4041                            |
          |                                            |
          |SPEAKER:  Dr. R. Wilson                     |
          |          Department of Mathematics, Caltech|
          |                                            |
          |TITLE:    The minimum distance of some      |
          |          polynomial codes                  |
          |============================================|

                             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.