The need to transmit and store massive amounts of data reliably and without error is a vital part of modern communications systems. Error-correcting codes play a fundamental role in minimising data corruption caused by defects such as noise, interference, crosstalk and packet loss. This book provides an accessible introduction to the basic elements of algebraic codes, and discusses their use in a variety of applications. The author describes a range of important coding techniques, including Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. Throughout the book, mathematical theory is illustrated by reference to many practical examples. The book was first published in 2003 and is aimed at graduate students of electrical and computer engineering, and at practising engineers whose work involves communications or signal processing.

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Algebraic Codes for Data Transmission

A general framework for shift register-based signature analysis is presented, and a mathematical model for this framework-based on coding theory-is developed. There are two key features of this formulation, first, it allows for uniform treatment of LFSR, MISR, and multiple MISR-based signature analyzer. In addition, using this formulation, a new compression scheme for multiple output CUT is proposed. This scheme, referred to as multiinput LFSR, has the potential to achieve better aliasing than other schemes such as the multiple MISR scheme of comparable hardware complexity. Several results on aliasing are presented, and certain known results are shown to be direct consequences of the formulation. Also developed are error models that take into account the circuit topology and the effect of faults at the outputs. Using these models, exact closed-form expressions for aliasing probability are developed. A closed-form aliasing expression for MISR under an independent error model is provided. >

A new framework for designing and analyzing BIST techniques and zero aliasing compression

Using algebraic geometry codes we give a polynomial construction of quantum codes with asymptotically nonzero rate and relative distance.

Asymptotically Good Quantum Codes

Single and multiple multiple-input-signature-register (MISR) aliasing probability expressions are presented for arbitrary test lengths. A framework, based on algebraic codes, is developed for the analysis and synthesis of MISR-based test response compressors for BIST. This framework is used to develop closed-form expressions for the aliasing probability of MISR for arbitrary test length. An error model, based on q-ary symmetric channel, is proposed using more realistic assumptions. Results are presented that provide the weight distributions for q-ary codes (q=2/sup m/, where the circuit under test has m outputs). These results are used to compute the aliasing probability for the MISR compression technique for arbitrary test lengths. This result is extended to compression using two different MISRs. It is shown that significant improvements can be obtained by using two signature analyzers instead of one. The weight distribution of a class of codes of arbitrary length is also given. >

Aliasing probability for multiple input signature analyzer

We propose a method for detecting the generation of garbage cells by analyzing a source text written in a functional programming language which uses ordinary linked lists to implement list-type values. For a subexpression such as F(G( . . . )) in a program where the function values of F and G are of list type, if a cell c is created during the computation of G and if c does not appear in a list-type value of F, then c becomes a garbage cell at the end of the computation of F. We discuss this problem on the basis of formal languages derived from the functional program text and show some sufficient conditions that predict the generation of garbage cells. Also, we give an efficient algorithm to detect at compile time the generation of garbage cells which are linearly linked. We have implemented these algorithms in an experimental LISP system. By executing several sample programs on the system, we conclude that our method is effective in detecting the generation of garbage cells.

Analysis of functional programs to detect run-time garbage cells

The binary weight distribution of the extended (2m,2m − 4) code of the Reed-Solomon code over GF(2m) with generator polynomial (x − α)(x − α2)(x − α3)

Functional languages are known to have such advantages as simply defined semantics. However little study has been made regarding compilation or optimization methods for efficient execution on a conventional machine. This paper formulates some optimization methods compiling a functional language ASL/F program into an object program in a procedural language, and further gives some sufficient conditions to perform these optimizations. It introduces “needed-argument-first computation” and “globalization of sorts (data types),” which are new optimization methods and have not been discussed before. Sample programs which are compiled and executed by an experimental compiler perform these optimization methods, as well as elimination of duplicated computation for common subterms, elimination of tail recursion, and rewriting in compilation. These experimental results show that execution time and dynamic memory space requirement of the object program are reduced by these optimizations and the program can be executed in almost the same time as an equivalent program written in a procedural language such as PASCAL.

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Optimization of functional language ASL/F programs

An algebraic specification (or text) specifies a congruence relation on a set of expressions. In algebraic language $ASL/*$ , a pair $(G, AX)$ is called a text, where $G$ is a context-free grammar and $AX$ is a set of axioms. A text $t=(G, AX)$ specifies the set $E_{G}$ of expressions generated by $G$ and the least congruence relation on $E_{G}$ satisfying all the axioms in $AX$ , where ‘congruency’ is defined based on the syntax (phrase structure) of the expressions. In general, for a text $t$ in $ASL/*$ , the condition, (A) $e$ is congruent with $e’$ in $t$ , is not logically equivalent to the condition, (B) $e’$ is obtained from $e$ by rewriting $e$ when the axioms in $t$ are regarded as ‘bidirectional‘ rewrite rules. We present a sufficient condition for a text $t$ under which (A) and (B) are equivalent for any pair of expressions $e$ and $e’$ , which means that the congruence relation specified by $t$ is simply defined operationally.

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Tadao Kasami

Papers

The binary weight distribution of the extended (2m,2m − 4) code of the Reed-Solomon code over GF(2m) with generator polynomial (x − α)(x − α2)(x − α3)

Optimization of functional language ASL/F programs

On Operational Semantics of Congruence Relation Defined in Algebraic Language ASL/$^\ast$