Abstract: Data are received that represents current prices of options on a given asset. An estimate is derived from the data of a corresponding implied probability distribution of the price of the asset at a future time. Information about the probability distribution is made available within a time frame that is useful to investors, for example, promptly after the current option price information becomes available.

Abstract: Data are received that represents current prices of options on a given asset. An estimate is derived from the data of a corresponding implied probability distribution of the price of the asset at a future time. Information about the probability distribution is made available within a time frame that is useful to investors, for example, promptly after the current option price information becomes available.

Abstract: Data are received that represents current prices of options on a given asset. An estimate is derived from the data of a corresponding implied probability distribution of the price of the asset at a future time. Information about the probability distribution is made available within a time frame that is useful to investors, for example, promptly after the current option price information becomes available.

Abstract: Data are received that represents current prices of options on a given asset. An estimate is derived from the data of a corresponding implied probability distribution of the price of the asset at a future time. Information about the probability distribution is made available within a time frame that is useful to investors, for example, promptly after the current option price information becomes available.

Abstract: Data are received that represents current prices of options on a given asset. An estimate is derived from the data of a corresponding implied probability distribution of the price of the asset at a future time. Information about the probability distribution is made available within a time frame that is useful to investors, for example, promptly after the current option price information becomes available.

Abstract: Vectors are quantized by representing the vectors by quantized values using a codebook having granular regions which are based on a coset code, and which lie inside a boundary region which is based on another coset code, the boundary region being other than an N-cube and other than the Voronoi region of a sublattice .LAMBDA..sub.b =M.LAMBDA..sub.g of a lattic .LAMBDA..sub.g upon which the granular regions may have been based.

Abstract: In general the invention features mapping a digital data sequence into a signal point sequence for data transmission over a channel characterized by a non-ideal response, by selecting the signal point sequence from a subset of all possible signal point sequences based on the digital data sequence and upon the response, all possible signal point sequences in the subset lying in a fundamental region of a filtered trellis code, the fundamental region being other than a simple Cartesian product of finite-dimensional regions.

Abstract: A digital data sequence is mapped into a signal point sequence for transmission, by selecting the signal point sequence from a subset of all possible signal point sequences based on the digital data sequence, all possible signal point sequences in the subset lying in a fundamental region of a trellis code, the fundamental region being other than a simple Cartesian product of finite-dimensional regions. In another aspect, a digital data sequence is mapped into a sequence of signal points for transmission, by specifying a class of possible sequences based on the digital data, and selecting the signal point sequence from the class, the selection being based on the respective average powers of the possible sequences of the class, the selection being based on not only a fixed-length block of the digital data.

Abstract: A Voronoi signal constellation with ties includes those points of a lattice .LAMBDA. (or a coset .LAMBDA.+c) that lie within a Voronoi region of a sublattice .LAMBDA.' of .LAMBDA., .LAMBDA.' being other then a scaled version of .LAMBDA., and the constellation includes more than .vertline..LAMBDA./.LAMBDA.'.vertline. points, where .vertline..LAMBDA./.LAMBDA.'.vertline. is the order of the lattice partition .LAMBDA./.LAMBDA.'. In order aspects, the lattices .LAMBDA. and .LAMBDA.' are of dimension greater than two; the sublattice .LAMBDA.' is a version of a binary lattice of depth at least two and normalized informativity less than one; a method is provided of mapping from m data bits to a point drawn from a Voronoi constellation of more than 2.sup.m points, based on an N-dimensional lattice partition .LAMBDA./.LAMBDA.'; and Voronoi constellations are provided comprising points of a lattice .LAMBDA. (or a coset .LAMBDA.+c) that lie within a Voronoi region of a sublattice .LAMBDA.' of .LAMBDA., the sublattice .