Information-theoretic approach to unimodal density estimation

Ex = (I + y / t) (l +. r y / : ') El0 = (1 + ! / / Z) (l + .r2y/:3) all R-multiples of Es. The error positions in this case are (j. 1. l) , In general, when decoding an error relative to an algebraic geometric code C*(D. 7rtP), there is a vector space S (g) l of error-locator functions of dimension Z(m)-e. Most algorithms settle for any element of this space as an error-locator and deal with extraneous zeros later. If one considers the ideal Z of all error-locator functions, then there is a generating set of size at most p, the smallest positive pole size at P. The one-dimensional BerlekampMassey version of the Feng-Rao algorithm given here is sufficient to reasonably efficiently produce such a generating set, and the error positions (for any error of weight at most c < i6:t, S " *, = m-29 + 2, the designed distance of the code) will be exactly the common zeros L'(2) of those error-locator functions. (For further efficiency, " higher dimensional " BerlekampMassey algorithms can be worked out in a straightforward manner as well.) This Feng-Rao type algorithm gives the designed distance independent of the Riemann-Roch theorem, and the algorithm to get these is merely row-reduction with shifting (as with any BerlekampMassey type algorithm), coupled with a Feng-Rao majority-vote scheme to produce further syndromes. Moreover, such a strategy can be used on arbitrary divisors (5 (though at present it is not provable that one can achieve decoding up to the designed distance efficiently in this manner). The generating set found may, in addition, allow for efficient calculation of the common zeros. So the algorithm given here has the advantages that 1) it treats all projective points, 2) it decodes up to the designed minimum distance, 3) it uses a (one-dimensional) Berlekamp-Massey row-reduction algorithm to efficiently (that is with roughly what should be expected as a running time) row-reduce S , 4) it produces a small set of generators for the whole error-locator ideal Z, rather than settling for a single error-locator function with possibly extraneous zeros. (A minimal Grobner basis can be extracted from this or produced directly from a BerlekampMassey type row-reduction algorithm that treats rows of the syndrome matrix as grids generated by the minimal nonzero elements of C(@). and shifts in all the grid directions.) Decoding algebraic-geometric codes up to …