Generalized Reed-Muller Research Articles

New Locally Correctable Codes Based on Projective Reed–Muller Codes

Locally decodable codes and locally correctable codes (LCCs) have several important applications, such as private information retrieval, secure multiparty computation, and circuit lower bounds. Three major parameters are considered in LCCs: query complexity, message length, and codeword length. The most familiar LCCs in the regime of low query complexity are the generalized Reed–Muller (GRM) codes. However, it has not previously been determined whether there exist codes that have shorter codeword lengths than GRM codes with the same query complexity and message length. In this paper, we show that the projective Reed–Muller (PRM) codes are such LCCs for some parameters. The GRM code is specified by the alphabet size $q$ , the number of variables $m$ , and the degree $d$ , where $d\le q-2$ . When $d= q-2$ and $q-1$ is a power of a prime, we prove that there exists a PRM code with shorter codeword length than the GRM code with the same query complexity and message length. We also present for these PRM codes a perfectly smooth local decoder to recover a symbol in a codeword by accessing not more than $q$ symbols at the coordinates of the codeword.

On low weight codewords of generalized affine and projective Reed–Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed---Muller (GRM) codes. In the affine case we prove that if the cardinality of the ground field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then, without this assumption on the cardinality of the field, we study codewords associated to an irreducible but not absolutely irreducible polynomial, and prove that they cannot be second, third or fourth weight depending on the hypothesis. In the projective case the second distance of GRM codes is estimated, namely a lower bound and an upper bound on this weight are given.

NEW CONSTRUCTION OF GRM CODES

In this paper we study a new class of generalized Reed–Muller (GRM) codes with respect to a new construction that involves multiple GRM codes. We have proved that the resultant codes are also GRM codes.

Generalized Reed–Muller codes over $${\mathbb{Z}_q}$$

We have given a generalization of Reed---Muller codes over the prime power integer residue ring $${\mathbb{Z}_q}$$ . These codes are analogs of generalized Reed---Muller (GRM) codes over finite fields. We mainly focus on primitive GRM codes, which are basically a generalization of Quaternary Reed---Muller (QRM) codes. We have also given a multivariate representation of these codes. Non-primitive GRM codes over $${\mathbb{Z}_q}$$ are also briefly discussed. It has been shown that GRM codes over $${\mathbb{Z}_q}$$ are free extended cyclic codes. A trace description of these codes is also given. We have obtained formulas for their ranks and also obtained expressions for their minimum Hamming distances.

Testable design of AND–EXOR logic networks with universal test sets

This paper presents a testability enhancement technique suited for AND–EXOR based logic networks that facilitates easy detection of stuck-at and bridging faults by a universal test set. Both cascaded and tree implementations of the EXOR-part are considered. The AND–EXOR based circuit implemented with a cascaded EXOR-part requires a universal test set of size (2 n + 6) for an n-variable function implementation. For Generalized Reed–Muller (GRM) implementation, this test set detects all single stuck-at and bridging faults (both OR-type and AND-type) and also large number of multiple bridging faults. For an Exclusive-OR Sum-of-Products (ESOP) implementation, a few single bridging faults may remain untested under this test set, occurrence of which can be minimized by employing an appropriate design and layout technique. Next, it is shown that an AND–EXOR network with a tree-based EXOR-part can be tested for similar faults by a universal test set of size (2 n + 8). This paper also solves an open problem of designing a universal test for a tree-based AND–EXOR circuit. Since the EXOR-tree has depth of ( ⌈ log 2 s ⌉ + 1 ), where s is the number of product terms in the given AND–EXOR expression, this tree-based design reduces the circuit delay significantly compared to cascaded EXOR implementation. In both the cases, the test set can be stored in a ROM on-chip for built-in self-test (BIST) purposes. For several benchmark circuits, the universal test set is found to be much smaller in size than the ATPG-generated test sets.

Testing Polynomials over General Fields

In this work we fill the knowledge gap concerning testing polynomials over finite fields. As previous works show, when the cardinality of the field, $q$, is sufficiently larger than the degree bound, $d$, then the number of queries sufficient for testing is polynomial or even linear in $d$. On the other hand, when $q=2$ then the number of queries, both sufficient and necessary, grows exponentially with $d$. Here we study the intermediate case where $2 < q = O(d)$ and show a smooth transition between the two extremes. Specifically, let $p$ be the characteristic of the field (so that $p$ is prime and $q = p^s$ for some integer $s \geq 1$). Then the number of queries performed by the test grows like $\ell\cdot q^{2\ell+1}$, where $\ell = \big\lceil \frac{d+1}{q-q/p}\big\rceil $. Furthermore, $q^{\Omega(\ell)}$ queries are necessary when $q = O(d)$. The test itself provides a unifying view of the tests for these two extremes: it considers random affine subspaces of dimension $\ell$ and verifies that the function restricted to the selected subspaces is a polynomial of degree at most $d$. Viewed in the context of coding theory, our result shows that Reed-Muller codes over general fields (usually referred to as generalized Reed-Muller (GRM) codes) are locally testable. In the course of our analysis we provide a characterization of small-weight words that span the code. Such a characterization was previously known only when the field size is a prime or is sufficiently large, in which case the minimum-weight words span the code.

Affine Invariant Extended Cyclic Codes Over Galois Rings

Recently, Blackford and Ray-Chaudhuri used transform domain techniques to permutation groups of cyclic codes over Galois rings. They used the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length 2/sup m/ over any subring of GR(4,m) to be affine invariant. Here, we use the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length p/sup m/ over any subring of GR(p/sup e/,m) to be affine invariant, for e=2 with arbitrary p and for p=2 with arbitrary e. These are used to find two new classes of affine invariant Bose-Chaudhuri-Hocquenghem (BCH) and generalized Reed-Muller (GRM) codes over Z/sub 2//sup e/ for arbitrary e and a class of affine invariant BCH codes over Z/sub p//sup 2/ for arbitrary prime p.

Testing of stuck-open faults in generalised Reed–Muller and EXOR sum-of-products CMOS circuits

Testable designs of GRM (generalised Reed–Muller) and ESOP (EXOR sum-of-products) circuits have been proposed for robustly detecting all single stuck-open faults in their CMOS implementation. It is shown that for an n-variable boolean function, a sequence of (4n + 13) vectors is sufficient to detect all single stuck-open faults in a GRM circuit. For an ESOP circuit, a test sequence of length (2n + 10) is sufficient. In the first case, the EXOR part is designed as a tree of depth ≤2(log⌈p + 1 ⌉), and for the latter, as a linear cascade of length ≤(p + 1), where p is the number of product terms in the GRM or ESOP expression. For both the cases the test sequence is universal, i.e. independent of the function and the circuit under test, and can be stored in a ROM for built-in self-test.

Subcodes of the Projective Generalized Reed-Muller Codes Spanned by Minimum-Weight Vectors

We use methods of Mortimer l19r to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length \frac{q^m - 1}{q-1} over the finite field Fq of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0<r<m−1 unless q is prime. The methods of proof are the projective analogue of those developed in l17r, and show that the codes spanned by the minimum-weight vectors are spanned over Fq by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1.

Baldwinian learning utilizing genetic and heuristic algorithms for logic synthesis and minimization of incompletely specified data with Generalized Reed–Muller (AND–EXOR) forms

Abstract This research applies a new heuristic combined with a genetic algorithm (GA) to the task of logic minimization for incompletely specified data, with both single and multi-outputs, using the Generalized Reed–Muller (GRM) equation form. The GRM equation type is a canonical expression of the Exclusive-Or Sum-of-Products (ESOPs) type, in which for every subset of input variables there exists not more than one term with arbitrary polarities of all variables. This AND–EXOR implementation has been shown to be economical, generally requiring fewer gates and connections than that of AND–OR logic. GRM logic is also highly testable, making it desirable for FPGA designs. The minimization results of this new algorithm tested on a number of binary benchmarks are given. This minimization algorithm utilizes a GA with a two-level fitness calculation, which combines human-designed heuristics with the evolutionary process, employing Baldwinian learning. In this algorithm, first a pure GA creates certain constraints for the selection of chromosomes, creating only genotypes (polarity vectors). The phenotypes (GRMs) are then learned in the environment and contribute to the GA fitness (which is the total number of terms of the best GRM for each output), providing indirect feedback as to the quality of the genotypes (polarity vectors) but the genotype chromosomes (polarity vectors) remain unchanged. In this process, the improvement in genotype chromosomes (polarity vectors) is the product of the evolutionary processes from the GA only. The environmental learning is achieved using a human-designed GRM minimization heuristic. As much previous research has presented the merit of AND–EXOR logic for its high density and testability, this research is the first application of the GRM (a canonical AND–EXOR form) to the minimization of incompletely specified data.

A Gap in GRM Code Weight Distributions

The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over \gf{3}. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3^{m-2} in \grm_3(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+\Delta, where d is the minimum distance and \Delta is the largest integer dividing all weights occurring in the code.

Cyclic subcodes of generalized Reed-Muller codes

We consider certain subcodes of generalized Reed-Muller (GRM) codes, which we call homogeneous generalized Reed-Muller (HRM) codes. In general, they have a much better minimum distance than the GRM codes. The parameters of HRM codes are related to those of projective Reed-Muller (PRM) codes. Unlike most PRM codes, punctured HRM codes are cyclic. Under the trace map, HRM codes map to binary codes. These are in general much larger than classical RM codes, for the same minimum distance.

Constructing Reed-Muller universal logic module networks from Reed-Muller binary decision diagrams

This paper presents the Reed-Muller binary decision diagram (RMBDD), an alternative means of representing generalized Reed-Muller (GRM) expansions. Particular emphasis is placed on the use of ordered Reed-Muller binary decision diagrams (ORMBDDs) as tools for constructing minimal (sub-minimal) Reed-Muller universal logic module (RM-ULM) networks. Initially, the structure of ordered Reed-Muller binary decision trees (ORMBDTs) is described and rules for reducing their size are introduced. The physical implementation of reduced ordered Reed-Muller binary decision diagrams (RORMBDDs) as RM-ULM networks is then illustrated. Additionally, a heuristic technique for determining the ordering of variables in an ORMBDD is detailed.

Generalized Reed-Muller forms as a tool to detect symmetries

In this paper, we present a new method for detecting groups of symmetric variables of completely specified Boolean functions. The canonical Generalized Reed-Muller (GRM) forms are used as a powerful analysis tool. To reduce the search space we have developed a set of signatures that allow us to identify quickly sets of potentially symmetric variables. Our approach allows for detecting symmetries of any number of inputs simultaneously. Totally symmetric functions can be detected very quickly. The traditional definitions of symmetry have also been extended to include more types. This extension has the advantage of grouping input variables into more classes. Experiments have been performed on MCNC benchmark cases and the results verify the efficiency of our method.

Minimisation of fixed-polarity AND/XOR canonical networks

A method for extracting the cubes of the generalised Reed-Muller (GRM) form of a Boolean function with a given polarity is presented. The method does not require exponential space and time complexity and it achieves the lower-bound time complexity. The proof of the method&apos;s correctness constitutes the first half of the paper. Also, a separate heuristic algorithm to find the optimal polarity that requires the least number of cubes in the GRM representation is proposed. The algorithm is fast and derives the polarity for variables and extracts all cubes simultaneously. It is based on the concept of a Boolean centre for vertices, which emulates the centre of gravity concept in geometry. The experimental results for the heuristic algorithm agree strongly with the authors observations and analysis.

Reed-Muller universal logic module networks

The paper describes Reed-Muller universal logic modules (RM-ULMs) and their use for the implementation of logic functions given in Reed-Muller (RM) form. A programmed algorithm is presented for the synthesis and optimisation of RM-ULM networks. The level-by-level minimisation procedure is based on the selection of control variables at different levels with the aim of maximising the number of discontinued branches and hence minimising the number of modules required to implement a given function. The algorithm is programmed in Fortran and can be used to realise fixed-polarity generalised Reed-Muller (GRM) expansions of any polarity and any number of variables.

Multiple valued input generalised reed—muller forms

The concept of canonical multiple valued input generalised Reed-Muller (MIGRM) forms is introduced. The MIGRM is a direct extension of the well known generalised Reed-Muller (GRM) forms to the logic with multiple valued inputs. The concept of the polarity of a GRM form is generalised to the polarity matrix of a MIGRM form. A tabular pattern-matching method is presented for the calculation of a MIGRM form. The MIGRM transform has been implemented for further investigations of such forms and their comparison with other circuit realisations.

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>

Efficient computer method for ExOR logic design

Application of exclusive-OR logic design suffers from a lack of straightforward design methods. Recently a procedure using generalised Reed-Muller (GRM) coefficient maps has been proposed. Based on this approach, an efficient computer method is developed for the generation of all 2n sets of GRM coefficients of an n-variable Boolean function. Along with the coefficients a metric may be calculated from which the minimum cost set according to some criterion may be selected. The method requires a storage of 2n bits and an average of 2n-1 + n/2 ExOR single-bit operations per set of GRM coefficients.