Some results on arithmetic codes of composite length

In this paper we present a new upper bound on the minimum distance of binary cyclic arithmetic codes of composite length. Two new classes of binary cyclic codes of composite length are introduced

Optimal Parallel Lexicographic Sorting using a Fine-Grained Decomposition

Though non-comparison based sorting techniques like radix sorting can be done with less work than conventional comparison-based methods, they are not used for long keys. This is because even though parallel radix sorting algorithms process the keys in parallel, the symbols in the keys are processed sequentially. In this report, we give an optimal algorithm for lexicographic sorting that can be used to sort n m-bit keys on an EREW model in Ө (log nlogm) time with Ө (mn) work . This algorithm is not only as fast as any optimal non-comparison based algorithm, but can also be executed with less work. We also use the proposed algorithm to show that if n Ө (log n) unsigned binary numbers can be sorted optimally on an EREW PRAM than n unsigned binary numbers of unrestricted length can be sorted optimally on an EREW PRAM

DUAL-MODE SEQUENTIAL LOGIC FOR FUNCTION INDEPENDENT FAULT-TESTING

This paper presents a method of using hardware redundancy to ease the problem of fault testing in sequential logic networks. Sequential logic networks are constructed using two kinds of dual-mode logic gates, one of which is specifically required to initialize a feedback loop to some logic value. Initially, it is shown that these networks can be tested for all single stuck-at-faults with six function-independent tests. Next, this method is generalized to detect large classes of multiple faults with six function-independent tests. In both cases, the network must have the proper number of extra inputs

Optimal Parallel Solutions to the Neighbor Localization Problem and Integer Sorting: A Fine Grained Approach

In this report, a fine-grained decomposition approach is used to obtain an optimal parallel solution to the Neighbor Localization Problem, which in turn is œ used to sort n θ(log n)-bit numbers optimally on an EREW model. The model of computation used is the EREW Reconfigurable PRAM (R-PRAM) that permits the use of “very small” processors. The main result of this report is a parallel EREW R-PRAM algorithm that sorts n θ(log n)-bit numbers in θ(log n) time with θ(n log n) “work”. The proposed algorithm is asymptotically optimal in time and efficiency. If a weaker variant of the R-PRAM (called the ISR-PRAM) is used, the efficiency suffers only a slight degradation

Efficient Maximum-Likelihood Soft-Decision Decoding of Linear Block Codes Using Algorithm A

In this report we present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here is to convert the decoding problem into a search problem through a graph which is a trellis for an equivalent code of the transmitted code. Algorithm A*, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to drastically reduce the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. Simulation results for the ( 48, 24) and the (72, 36) binary extended quadratic residue codes and the (128, 64) binary extended BCH code are given to substantiate the above claim

Generalized Finite-Geometry Codes

A technique is presented for constructing cyclic codes that retain many of the combinatorial properties of finite-geometry codes, but are often superior to geometry codes. It is shown that L-step orthogonalization is applicable to certain subclasses of these codes

An Optimum Symbol-by Symbol decoding rule for linear codes

A decoding rule is presented which minimizes the probability of symbol error over a time-discrete memoryless channel for any linear error-correcting code when the code words are equiprobable. The complexity of this rule varies inversely with code rate, making the technique particularly attractive to high rate codes. Examples are given for both block and convolutional codes

Designing Efficient Maximum-Likelihood Soft-Decision Decoding Algorithms for Linear Block Codes Using Algorithm A*

In this report we present a class of efficient maximum-likelihood soft-decision decoding algorithms for linear block codes. The approach used here is to convert the decoding problem into a search problem through a graph which is a trellis for an equivalent code of the transmitted code. Algorithm A*, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to drastically reduce the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. For example, simulation results for the (128,64) binary extended BCH code indicate that for most real channels the proposed decoding algorithm is at least fifteen orders of magnitude more efficient in time and in space than that proposed by Wolf. Simulation results for the (104, 52) binary extended quadratic residue code are also given. These simulation results indicate that the use of Algorithm A* for decoding has resulted not only in an efficient soft-decision decoding algorithm for hitherto intractable linear block codes, but an algorithm which is in fact optimal as well

Decoding by Sequential Code Reduction

A general decoding method for cyclic codes is presented which gives promise of substantially reducing the complexity of decoders at the cost of a modest increase in decoding time (or delay). Significant reductions in decoder complexity for binary cyclic finite-geometry codes are demonstrated, and two decoding options for the Golay code are presented

Some Results on the Weight Structure of Cyclic Codes of Composite Length

In this work we investigate the weight structure of cyclic codes of composite length n = n1n2, where n1 and n2 are relatively prime. The actual minimum distances of some classes of binary cyclic codes of composite length are derived. For other classes new lower bounds on the minimum distance are obtained. These new lower bounds improve on the BCH bound for a considerable number of binary cyclic codes