Programs and systems of recurrence equations may be represented as sets of actions which are to be executed subject to precedence constraints. In may cases, actions may be labelled by integral vectors in some iterations domains, and precedence constraints may be described by affine relations. A schedule for such a program is a function which assigns an execution data to each action. Knowledge of such a schedule allows one to estimate the intrinsic degree of parallelism of the program and to compile a parallel version for multiprocessor architectures or systolic arrays. This paper deals with the problem of finding closed form schedules as affine or piecewise affine functions of the iteration vector. An algorithm is presented which reduces the scheduling problem to a parametric linear program of small size, which can be readily solved by an efficient algorithm.

Some efficient solutions to the affine scheduling problem: I. One-dimensional time

Stream processing is a term that is used widely in the literature to describe a variety of systems. We present an overview of the historical development of stream processing and a detailed discussion of the different languages and techniques for programming with streams that can be found in the literature. This includes an analysis of dataflow, specialized functional and logic programming with streams, reactive systems, signal processing systems, and the use of streams in the design and verification of hardware.

A survey of stream processing

The parallelization of many algorithms can be obtained using space-time transformations which are applied on nested do-loops or on recurrence equations. In this paper, we analyze systems of linear recurrence equations, a generalization of uniform recurrence equations. The first part of the paper describes a method for finding automatically whether such a system can be scheduled by an affine timing function, independent of the size parameter of the algorithm. In the second part, we describe a powerful method that makes it possible to transform linear recurrences into uniform recurrence equations. Both parts rely on results on integral convex polyhedra. Our results are illustrated on the Gauss elimination algorithm and on the Gauss-Jordan diagonalization algorithm.

/pdf/the-mapping-of-linear-recurrence-equations-on-regular-arrays-4lm6r3p4dg.pdf

The mapping of linear recurrence equations on regular arrays

A structured parity-check matrix H is proposed, wherein H is an expansion of a base matrix Hb and wherein Hb comprises a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2, and a second part comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. The expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and the expansion uses paired submatrices for an even number of 1s in hb.

Method and apparatus for encoding and decoding data

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Contention-free interleavers have been recently shown to be suitable for parallel decoding of turbo codes. In this correspondence, it is shown that permutation polynomials generate maximum contention-free interleavers, i.e., every factor of the interleaver length becomes a possible degree of parallel processing of the decoder. Further, it is shown by computer simulations that turbo codes using these interleavers perform very well for the Third Generation Partnership Project (3GPP) standard

On maximum contention-free interleavers and permutation polynomials over integer rings

The design of the turbo code internal permutation is crucial regarding the minimum Hamming distance (MHD) and the related achievable asymptotic gain. After having set down the permutation issues in a synthetic way, this paper presents a generic model which allows us to obtain large MHDs for turbo codes and also offers large possibilities of natural parallelism in the decoder architecture.

Designing good permutations for turbo codes: towards a single model

Thanks to the probabilistic message passing performed between its component decoders, a turbo decoder is able to provide strong error correction close to the theoretical limit. However, the minimum Hamming distance (dmin) of a turbo code may not be sufficiently large to ensure large asymptotic gains at very low error rates (the so-called flattening effect). Increasing the dmin of a turbo code may involve using component encoders with a large number of states, devising more sophisticated internal permutations, or increasing the number of component encoders. This paper addresses the latter option and proposes a modified turbo code in which a fraction of the parity bits are encoded by a rate-1, third encoder. The result is a noticeably increased dmin, which improves turbo decoder performance at low error rates. Performance comparisons with turbo codes and serially concatenated convolutional codes are given.

Improving the distance properties of turbo codes using a third component code: 3D turbo codes - [transactions letters]

Thanks to the message passing principle, turbo decoding is able to provide strong error correction near the theoretical (Shannon) limit. However, the minimum Hamming distance (MHD) of a turbo code may not be sufficient to prevent a detrimental change in the error rate vs. signal to noise ratio curve, the so-called flattening. Increasing the MHD of a turbo code may involve using component encoders with a large number of states, devising more sophisticated internal permutations, or increasing the dimension of the turbo code, i.e. the number of component encoders. This paper addresses the latter option and proposes a modified turbo code, in which some of the parity bits stemming from the classical component encoders are encoded by a rate-1, third encoder. The result is a significantly increased MHD, which improves turbo decoder performance at low error rates, at the expense of a very small increase in complexity. In this paper, we compare the performance of the proposed turbo code with that of the DVB-RCS turbo code and the DVB-S2 LDPC code. Comparisons with more complex 16-state TCs are also reported.

https://portail.telecom-bretagne.eu/publi/public/download.jsp?id_publication=2917

Adding a Rate-1 Third Dimension to Turbo Codes

https://hal.inria.fr/inria-00075355/document

Computability of recurrence equations

We present new scheduling techniques for systems of affine recurrence equations. We show that it is possible to extend earlier results on affine scheduling to the case when each variable of the system is scheduled independently of the others by an affine timing-function. This new technique makes it possible to analyze systems of recurrence equations with variables in different index spaces, and multi-step systolic algorithms. We illustrate our method on dynamic programming, LU decomposition and 2D conv& lution.

https://hal.inria.fr/inria-00075354/document

Yannick Saouter

Papers

Designing good permutations for turbo codes: towards a single model

Improving the distance properties of turbo codes using a third component code: 3D turbo codes - [transactions letters]

Adding a Rate-1 Third Dimension to Turbo Codes

Computability of recurrence equations

Scheduling affine parameterized recurrences by means of variable dependent timing functions