2,836 research outputs found
p-Adic estimates of Hamming weights in Abelian codes over Galois rings
A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more
Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting
Type A affine shuffles are compared with riffle shuffles followed by a cut.
Although these probability measures on the symmetric group S_n are different,
they both satisfy a convolution property. Strong evidence is given that when
the underlying parameter satisfies , the induced measures on
conjugacy classes of the symmetric group coincide. This gives rise to
interesting combinatorics concerning the modular equidistribution by major
index of permutations in a given conjugacy class and with a given number of
cyclic descents. It is proved that the use of cuts does not speed up the
convergence rate of riffle shuffles to randomness. Generating functions for the
first pile size in patience sorting from decks with repeated values are
derived. This relates to random matrices.Comment: Galley version for J. Alg.; minor revisions in Sec.
Flag-symmetry of the poset of shuffles and a local action of the symmetric group
We show that the poset of shuffles introduced by Greene in 1988 is
flag-symmetric, and we describe a "local" permutation action of the symmetric
group on the maximal chains which is closely related to the flag symmetric
function of the poset. A key tool is provided by a new labeling of the maximal
chains of a poset of shuffles, which is also used to give bijective proofs of
enumerative properties originally obtained by Greene. In addition we define a
monoid of multiplicative functions on all posets of shuffles and describe this
monoid in terms of a new operation on power series in two variables.Comment: 34 pages, 6 figure
On Bijective Variants of the Burrows-Wheeler Transform
The sort transform (ST) is a modification of the Burrows-Wheeler transform
(BWT). Both transformations map an arbitrary word of length n to a pair
consisting of a word of length n and an index between 1 and n. The BWT sorts
all rotation conjugates of the input word, whereas the ST of order k only uses
the first k letters for sorting all such conjugates. If two conjugates start
with the same prefix of length k, then the indices of the rotations are used
for tie-breaking. Both transforms output the sequence of the last letters of
the sorted list and the index of the input within the sorted list. In this
paper, we discuss a bijective variant of the BWT (due to Scott), proving its
correctness and relations to other results due to Gessel and Reutenauer (1993)
and Crochemore, Desarmenien, and Perrin (2005). Further, we present a novel
bijective variant of the ST.Comment: 15 pages, presented at the Prague Stringology Conference 2009 (PSC
2009
Central limit theorems for patterns in multiset permutations and set partitions
We use the recently developed method of weighted dependency graphs to prove
central limit theorems for the number of occurrences of any fixed pattern in
multiset permutations and in set partitions. This generalizes results for
patterns of size 2 in both settings, obtained by Canfield, Janson and
Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.Comment: version 2 (52 pages) implements referee's suggestions and uses
journal layou
p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)
Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support
- …