Reed-Muller Coefficients Research Articles

Mapping of on-set fixed polarity Reed-Muller coefficients from on-set canonical sum of products coefficients and the minimization of pseudo Reed-Muller expressions

In this paper, a more efficient tabular method for mapping on-set fixed polarity Reed–Muller coefficients for a given polarity vector from on-set canonical sum of products coefficients is developed. Using this mapping technique, a heuristic algorithm for generating optimal pseudo Reed–Muller expressions from canonical sum of products expressions is also developed. Illustrative examples and experimental results are included to show the performance of the developed algorithms.

Mapping of fixed polarity Reed-Muller coefficients from minterms and the minimization of fixed polarity Reed-Muller expressions

In this paper, a relationship between on-set minterms of a logic function, an arbitrary polarity vector of the variables, and the corresponding fixed polarity Reed-Muller (FPRM) coefficients is established. Using this relationship, an efficient and simple algorithm for mapping FPRM coefficients from the on-set minterms of the function for a given polarity vector is presented. Another heuristic algorithm for finding an optimal polarity vector from the on-set minterms that produces the near minimum fixed polarity Reed-Muller expression (FPRME) is also presented. Both these algorithms are developed for single-output fully specified logic functions.

Algorithms for conversion of minterms to positive polarity Reed-Muller coefficients and vice versa

In this paper, an algorithm for mapping positive polarity Reed-Muller (PPRM) coefficients from canonical sum of products (CSOP) coefficients is presented. Another algorithm for mapping CSOP coefficients is also presented. Both algorithms are more space efficient than the existing algorithms.

Simple disjoint decomposition of Reed-Muller polynomials

The theory for simple disjoint decomposition of Reed-Muller polynomials is presented. Decomposition charts rearranged from Reed-Muller coefficient maps are introduced as an alternative in finding simple disjoint decompositions. The use of Reed-Muller coefficients to find simple disjoint decompositions appears to be simpler than using minterms.

Test-pattern generation based on Reed-Muller coefficients

Reed-Muller coefficients are used to generate a test-pattern selection procedure for detecting single stuck-at faults. This procedure is based on the heuristics deduced from the way in which the spectral coefficients are affected by such faults. The changes that the spectral coefficients undergo are also compared with the fault models used most frequently to model defects in combinational circuits. The proposed pattern-generating procedure does not need a simulation of the circuit for each of the possible stuck-at faults, and its complexity is proportional to the number of gates in the circuit. The proportionality constant increases exponentially as the number of inputs in the circuit increases. To evaluate the performance of the proposed method, its application to some benchmark circuits, including the ALU 74181, is presented. >

Generalized Hopfield neural network for concurrent testing

The use of generalized Hopfield neural networks in designing the checking circuitry of a concurrent testable circuit is discussed. The aliasing probability, a measure for evaluating the performance of the checking circuitry, is provided. It is shown how, by using spectral techniques based on the Reed-Muller transform, the aliasing probability can be expressed as a function of the Reed-Muller coefficients. Therefore, obtaining the checking circuitry means selecting a set of Reed-Muller spectral coefficients, with fewer elements than a given bound, that minimizes the aliasing probability. To apply the neural networks to design the checking circuitry for concurrent testing, the aliasing probability has been used as an energy function, and the Hopfield neural network has been modified to have an associated energy function with any type of polynomial dependence on the processor states.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

One more way to calculate generalized Reed-Muller expansions of boolean functions

A new algorithm is given that converts disjoint cube representation of boolean functions into fixed-polarity generalized Reed-Muller expansions (GRME). Since the known fast algorithm that generates the GRME based on the factorization of the Reed-Muller transform matrix always starts from the truth table (minterms) of a boolean function, the method described has the advantages that come from requiring a smaller computer memory. Moreover, for the boolean functions described by only a few disjoint cubes, the method is much more efficient than the fast algorithm. The algorithm allows either the calculation of only selected Reed-Muller coefficients, or all the coefficients can be calculated in parallel.

An efficient method of computing generalized Reed-Muller expansions from binary decision diagram

An efficient method for the generation of all the 2/sup n/ sets of generalized Reed-Muller (GRM) coefficients for a Boolean function f(X) of n variables using the binary decision diagram (BDD) is presented. The author describes the generation of RM coefficients from minterm values and relates them to the associated subfunctions. Examples are included to illustrate the procedure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Invited paper. Boolean matrix representation for the conversion of minterms to Reed–Muller coefficients and the minimization of Exclusive-OR switching functions

Two new efficient algorithms are proposed to convert the minterms of a switching function to the coefficients of its Reed-Muller polynomial with fixed polarities. The conversion procedure is based on the use of boolean matrix representation. The first algorithm is applied to generate all the generalized Reed-Muller expansions of the related switching function and select the optimum solution from them. The second algorithm is applied to generate directly the minimal generalized Reed-Muller expansion for a given switching function without undertaking all possible fixed polarity realizations. The two algorithms can also be applied to generate the minimal Exclusive-OR realization in fixed polarity for the incompletely specified functions. Two fast and efficient computer simulations for the proposed algorithms have been developed using BASIC programming language, these computer programs accept the number of the function variables and the minterms of a switching function and return the minimal Reed-Muller expan...

Some results on Reed-Muller coefficients

A method for defining the Reed-Muller (RM) coefficients of the AND, OR, exclusive-OR and complementary boolean functions directly from the individual RM coefficients is presented. Some logical operations on RM coefficients are also discussed. These results may be useful for detecting faults in the RM expansion.

Graphical method for the conversion of minterms to Reed-Muller coefficients and the minimisation of exclusive-OR switching functions

Graphical method for the conversion of minterms to Reed-Muller coefficients and the minimisation of exclusive-OR switching functions

Mapping of Reed-Muller coefficients and the minimisation of exclusive OR-switching functions

A new geometric format, the bj coefficient map, is here introduced, the entries in which are the {0, l} coefficient values of the Reed-Muller Exclusive-OR expansions by which any given combinatorial function may be expressed. Although similar in format to the classic Karnaugh map, the bj map entries do not represent the function output in the same manner as do the minterm entries plotted on a Karnaugh map. It is shown that this coefficient map structure may readily be used to generate any required Exclusive-OR realisation of a given function, and provides a deeper insight into the coefficient relationships which arise when input variables are complemented in any Exclusive-OR expansion.