Johan Sebastian Rosenkilde Nielsen

A Behavioral Synthesis Frontend to the Haste/TiDE Design Flow

This paper presents a complete design tool which performs automatic behavioral synthesis of asynchronous circuits (resource sharing, scheduling and binding).The tool targets a traditional control-datapath-style template architecture. Within the limitations set by this template architecture it is possible to optimize for area (which is our main focus) or for speed. This is done by simply using different cost functions.Input to the tool is a behavioral description in the Haste language, and output from the tool is a Haste program describing the synthesized implementation consisting of adatapath and a controller. The tool may be seen as an add-on to the Haste/TiDE tool flow, and it can be used to automatically optimize parts of a design and to quickly explore alternative optimizations. The paper outlines the design flow, explains key elements of the design tool, and presents a number of benchmark results.

On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes

We derive the Wu list-decoding algorithm for generalized Reed-Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm as the initial algorithm instead of the Berlekamp-Massey algorithm. We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm, immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.

Generalised Multi-sequence Shift-Register synthesis using module minimisation

We show how to solve a generalised version of the Multi-sequence Linear Feedback Shift-Register (MLFSR) problem using minimisation of free modules over F[x]. We show how two existing algorithms for minimising such modules run particularly fast on these instances. Furthermore, we show how one of them can be made even faster for our use. With our modelling of the problem, classical algebraic results tremendously simplify arguing about the algorithms. For the non-generalised MLFSR, these algorithms are as fast as what is currently known. We then use our generalised MLFSR to give a new fast decoding algorithm for Reed Solomon codes.

Multi-trial Guruswami---Sudan decoding for generalised Reed---Solomon codes

An iterated refinement procedure for the Guruswami–Sudan list decoding algorithm for Generalised Reed–Solomon codes based on Alekhnovich’s module minimisation is proposed. The method is parametrisable and allows variants of the usual list decoding approach. In particular, finding the list of closest codewords within an intermediate radius can be performed with improved average-case complexity while retaining the worst-case complexity. We provide a detailed description of the module minimisation, reanalysing the Mulders–Storjohann algorithm and drawing new connections to both Alekhnovich’s algorithm and Lee–O’Sullivan’s. Furthermore, we show how to incorporate the re-encoding technique of Kötter and Vardy into our iterative algorithm.

Algorithms for Simultaneous Padé Approximations

We describe how to solve simultaneous Padé approximations over a power series ring K[[x]] for a field K using O~(nω - 1 d) operations in K, where d is the sought precision and

Sub-Quadratic Decoding of One-Point Hermitian Codes

n

On decoding Interleaved Chinese Remainder codes

is the number of power series to approximate. We develop two algorithms using different approaches. Both algorithms return a reduced sub-bases that generates the complete set of solutions to the input approximations problem that satisfy the given degree constraints. Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite Padé approximations.

Power Decoding of Reed–Solomon Codes Revisited

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami-Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the matrix minimization algorithms from computer algebra, yielding similar asymptotic complexities.

Decoding Interleaved Gabidulin Codes using Alekhnovich's Algorithm

We model the decoding of Interleaved Chinese Remainder codes as that of finding a short vector in a Z-lattice. Using the LLL algorithm, we obtain an efficient decoding algorithm, correcting errors beyond the unique decoding bound and having nearly linear complexity. The algorithm can fail with a probability dependent on the number of errors, and we give an upper bound for this. Simulation results indicate that the bound is close to the truth. We apply the proposed decoding algorithm for decoding a single CR code using the idea of “Power” decoding, suggested for Reed-Solomon codes. A combination of these two methods can be used to decode low-rate Interleaved Chinese Remainder codes.

Row reduction applied to decoding of rank-metric and subspace codes

Power decoding, or “decoding by virtual interleaving”, of Reed–Solomon codes is a method for unique decoding beyond half the minimum distance. We give a new variant of the Power decoding scheme, building upon the key equation of Gao. We show various interesting properties such as behavioural equivalence to the classical scheme using syndromes, as well as a new bound on the failure probability when the powering degree is 3.