Reed-Muller Research Articles

The quantum query complexity of learning multilinear polynomials

In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field Fq. Any bounded-error classical algorithm for this task requires Ω(nd) queries to the polynomial. We give an exact quantum algorithm that uses O(nd−1) queries for constant d, which is optimal. In the case q=2, this gives a quantum algorithm that uses O(nd−1) queries to identify a codeword picked from the binary Reed–Muller code of order d.

FHT를 이용한 RM부호 인식 알고리즘

The use of an error-correcting code is essential in digital communication systems where the channel is noisy. Unless a receiver has accurate channel coding parameters, it becomes difficult to decode the digitized encoding bits correctly. In this paper, estimation algorithm for RM(Reed-Muller) codes using FHT (Fast Hadamard algorithm) is proposed. The proposed algorithm estimates the channel coding parameters of RM codes and then decodes the codes using the characteristic of FHT. And we also verify the algorithm by performing intensive computer simulation in additive white gaussian noise (AWGN) channel.

Conditionally Cycle-Free Graphical Models for Coset Codes

Conditionally cycle-free graphical models (i.e., cyclic graphical models which become cycle-free after conditioning on a subset of the hidden variables) are constructed for coset codes. Following the description of a general construction procedure, examples of a number of families of codes-including first-order Reed-Muller (RM) and the Delsarte-Goethals codes - are provided for which the proposed procedure yields optimal soft-in soft-out (SISO) decoding algorithms that are less complex than the best known trellis-based algorithms. In the case of the first-order RM codes, which have a recursive coset construction, the optimal SISO decoding algorithm that results when the proposed construction is applied repeatedly is denoted recursive coset representation (RCR) decoding. Connections are made between RCR decoding and existing algorithms that exploit fast Hadamard transforms. Finally, the utility of the proposed decoding algorithms are supported by a practically motivated application: the construction of serially concatenated codes that have high rates and low error floors. Extended Hamming codes are proposed as outer codes in such constructions with efficient decoding employing the SISO algorithms developed herein.

A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes

We give a new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes published in 1970. To prove this theorem, we consider the intersection of the support of minimal weight codewords with affine hyperplanes and we proceed by recursion.

Security Enhanced Key Predistribution Scheme Using Transversal Designs and Reed Muller Codes for Wireless Sensor Networks

Resource constraints of the nodes make security protocols difficult to implement. Thus key management is an important area of research in Wireless Sensor Networks (WSN). Key predistribution (kpd) which involves preloading keys in sensor nodes, has been considered as the best solution for key management when sensor nodes are battery powered and have to work unattended. This paper proposes a method to fix some loophole in an existing key predistribution scheme thereby enhancing the security of messages exchanged within a WSN. Here we use a model based on Reed Muller Codes to establish connectivity keys between sensor nodes. The model is then utilized to securely establish communication keys and exchange messages in a WSN designed on basis of two schemes using transversal designs for key predistribution. The combination of the key predistribution scheme and the connectivity model gives rise to highly resilient communication model with same connectivity between nodes as the chosen key predistribution scheme.

Novel FEC Coding Scheme for Dimmable Visible Light Communication Based on the Modified Reed–Muller Codes

In this letter, we propose a novel forward error correction (FEC) coding method using a modified Reed-Muller (RM) code for providing accurate dimming control in on-off keying (OOK) based visible light communication (VLC) systems. The proposed FEC scheme has a property that produces exactly same occurrence probability of 1's and 0's. Therefore, it can contain minimal compensation symbols in supporting multiple dimming target values. Moreover, VLC with our proposed scheme provides more coding gain than VLC scheme with the conventional FEC codes [the first-order RM and Reed-Solomon (RS) codes]. Simulation results for various dimming target values show that the proposed scheme leads to at least 2 dB or more signal-to-noise ratio (SNR) gain in 1% bit error rate (BER) than two FEC codes.

Highly Secure Key Predistribution Using Affine Planes And Reed Muller Codes In Wireless Sensor Networks

Wireless Sensor Networks (WSN) consist of low powered and resource constrained sensor nodes which are left unattended for long duration of time. Hence it is very challenging to design and implement cost effective security protocols for such networks. Thus symmetric key cryptographic techniques are preferred over public key techniques for communication in such scenarios. Prior to deployment, keys are usually predistributed into the nodes and this problem has been well studied. Highlighting that connectivity and communication are two separate aspects of a WSN, we propose a secure connectivity model using Reed Muller codes. The model is then utilized to securely establish communication keys and exchange messages in a WSN designed on the basis of a scheme that uses affine planes for key predistribution. By the introduction of connectivity model, the node identifiers (ids) are converted from public to private information to each node. These private node ids can be used to generate new communication keys from old ones by applying cryptographic hash functions. Novel combination of these ideas yields highly resilient communication model with full connectivity between nodes.

Stopping Set Distributions of Some Reed–Muller Codes

Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> be a binary [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ] linear code with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> , where the rows of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> may be dependent. A stopping set <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is a subset of column indices of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> such that the restriction of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> does not contain a row of weight one. The stopping set distribution { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> )} <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =0 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> enumerates the number of stopping sets with size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> . Note that stopping sets and stopping set distribution are related to the parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> . Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> be the parity-check matrix of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> which is formed by all the nonzero codewords of its dual code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> . A parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is called BEC-optimal if <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> )= <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> ), <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =0,1,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.

Bent and Hyper-Bent Functions in Polynomial Form and Their Link With Some Exponential Sums and Dickson Polynomials

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials) and give some conjectures that lead to constructions of new hyper-bent functions.

Classification of Some Families of Quaternary Reed–Muller Codes

Recently, new families of quaternary linear Reed–Muller codes have been introduced. They satisfy that, after the Gray map, the corresponding <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{4}$</tex></formula> -linear codes have the same parameters and properties as the codes of the binary linear Reed–Muller family. A structural invariant, the dimension of the kernel, for binary codes is used to classify completely these <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\BBZ}_{4}$</tex></formula> -linear codes. The dimension of the kernel for these <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{4}$</tex> </formula> -linear codes is established generalizing the known results about the dimension of the kernel for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{4}$</tex> </formula> -linear Hadamard and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{4}$</tex> </formula> -linear extended 1-perfect codes.

Weight enumeration of codes from finite spaces

We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a finite projective or affine space. As a result from the geometric method we use for the weight enumeration, we also completely determine the set of supports of subcodes and words in an extension code.

Random low-degree polynomials are hard to approximate

We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over $${\mathbb F}$$. We prove that almost all degree d polynomials have only an exponentially small correlation with all polynomials of degree at most d − 1, for all degrees d up to Θ(n). That is, a random degree d polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low-degree polynomial. Recently, several results regarding the weight distribution of Reed–Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed–Muller codes.

On the non-minimality of the largest weight codewords in the binary Reed-Muller codes

The study of minimal codewords in linear codes was motivated by Massey who described how minimal codewords of a linear code define access structures for secret sharing schemes. As a consequence of his article, Borissov, Manev, and Nikova initiated the study of minimal codewords in the binary Reed-Muller codes. They counted the number of non-minimal codewords of weight $2d$ in the binary Reed-Muller codes RM$(r,m)$, and also gave results on the non-minimality of codewords of large weight in the binary Reed-Muller codes RM$(r,m)$. The results of Borissov, Manev, and Nikova regarding the counting of the number of non-minimal codewords of small weight in RM$(r,m)$ were improved by Schillewaert, Storme, and Thas who counted the number of non-minimal codewords of weight smaller than $3d$ in RM$(r,m)$. This article now presents new results on the non-minimality of large weight codewords in RM$(r,m)$.

On Matrix Rigidity and Locally Self-correctable Codes

e describe a new approach for the problem of finding {\rm rigid} matrices, as posed by Valiant \cite{Val77}, by connecting it to the, seemingly unrelated, problem of proving lower bounds for linear locally self-correctable codes. This approach, if successful, could lead to a non-natural property (in the sense of Razborov and Rudich \cite{RR97}) implying super-linear lower bounds for linear functions in the model of logarithmic-depth arithmetic circuits. Our results are based on a lemma saying that, if the generating matrix of a locally decodable code is {\bf not} rigid, then it defines a locally self-correctable code with rate close to one. Thus, showing that such codes cannot exist will prove that the generating matrix of {\em any} locally decodable code (and in particular Reed Muller codes) is rigid.

Shared Autocorrelation Property of Sequences

Due to the low peak-to-mean envelope power ratio (PMEPR) of Golay sequences together with their connection to generalized Reed-Muller codes, many authors have proposed using Golay sequences as codewords in orthogonal frequency-division multiplexing (OFDM). Golay sequences that unexpectedly share the same aperiodic autocorrelation function can be used to construct longer Golay sequences. We study the shared autocorrelation property of general sequences with the ultimate goal of using those results to find other low power sequences. In this paper, we provide several new constructions of families of pairs of sequences with the shared autocorrelation property that enable us to explain nearly all of the binary shared autocorrelation property for sequences of lengths up to 24.

Reed-Muller function optimization techniques with onset table

By mapping a fixed polarity Reed-Muller (RM) expression into an onset table and studying the properties of the onset table, an algorithm is proposed to obtain a compact multi-level single-output mixed-polarity RM function by searching for and extracting the common variables using the onset table. Furthermore, by employing the multiplexer model, the algorithm is extended to optimize multi-level multi-output mixed-polarity RM forms. The proposed algorithm is implemented in C language and tested using some MCNC benchmarks. Experimental results show that the proposed algorithm can obtain a more compact RM form than that under fixed polarity. Compared with published results, the proposed algorithm makes a significant speed improvement, with a small increase in the number of literals.

Matrix Codes for Reliable and Cost Efficient Memory Chips

This paper presents a method to protect memories against multiple bit upsets and to improve manufacturing yield. The proposed method, called a Matrix code, combines Hamming and Parity codes to assure the improvement of reliability and yield of the memory chips in the presence of high defects and multiple bit-upsets. The method is evaluated using fault injection experiments. The results are compared to well-known techniques such as Reed-Muller and Hamming codes. The proposed technique performs better than the Hamming codes and achieves comparable performance with Reed-Muller codes with very favorable implementation gains such as 25% reduction in area and power consumption. It also achieves reliability increase by more than 50% in some cases. Further, the yield benefits provided by the proposed method, measured by the yield improvements per cost metric, is up to 300% better than the ones provided by Reed-Muller codes.

Efficient Deterministic Compressed Sensing for Images with Chirps and Reed–Muller Codes

A recent approach to compressed sensing using deterministic sensing matrices formed from discrete frequency-modulated chirps or from Reed–Muller codes is extended to support efficient deterministic reconstruction of signals that are much less sparse than envisioned in the original work. In particular, this allows the application of this approach in imaging. The reconstruction algorithm developed for images incorporates several new elements to improve computational complexity and reconstruction fidelity in this application regime.

Noisy Interpolating Sets for Low-Degree Polynomials

$\newcommand{\F}{{\mathbb F}}$ A Noisy Interpolating Set (NIS) for degree-$d$ polynomials is a set $S \subseteq \F^n$, where $\F$ is a finite field, such that any degree-$d$ polynomial $q \in \F[x_1,\ldots,x_n]$ can be efficiently interpolated from its values on $S$, even if an adversary corrupts a constant fraction of the values. In this paper we construct explicit NIS for every prime field $\F_p$ and any degree $d$. Our sets are of size $O(n^d)$ and have efficient interpolation algorithms that can recover $q$ from a fraction $\exp(-O(d))$ of errors. Our construction is based on a theorem which roughly states that if $S$ is a NIS for degree-1 polynomials then $d \cdot S= \{ a_1 + \ldots + a_d \,|\, a_i \in S\}$ is a NIS for degree-$d$ polynomials. Furthermore, given an efficient interpolation algorithm for $S$, we show how to use it in a black-box manner to build an efficient interpolation algorithm for $d \cdot S$. As a corollary we obtain an explicit family of punctured Reed-Muller codes (codes that are restrictions of a Reed-Muller code to a subset of the coordinates) which are good codes and have an efficient decoding algorithm against a constant fraction of errors. To the best of our knowledge, even the existence of punctured Reed-Muller codes that are good codes was not previously known.

Matching Vector Codes

An $(r,\delta,\epsilon)$-locally decodable code encodes a $k$-bit message $x$ to an $N$-bit codeword $C(x)$, such that for every $i\in[k]$, the $i$th message bit can be recovered with probability $1-\epsilon$, by a randomized decoding procedure that queries only $r$ bits, even if the codeword $C(x)$ is corrupted in up to $\delta N$ locations. Recently a new class of locally decodable codes (LDCs), based on families of vectors with restricted dot products, has been discovered. We refer to those codes as matching vector (MV) codes. Several families of $(r,\delta,\Theta(r\delta))$-locally decodable MV codes have been obtained. While codes in those families were shorter than codes of earlier generations, they suffered from having large values of $\epsilon=\Omega(r\delta)$, which meant that $r$-query MV codes could only handle error rates below $\frac{1}{r}$. Thus larger query complexity gave shorter length codes but at the price of less error tolerance. No MV codes of a superconstant number of queries capable of tolerating a constant fraction of errors were known to exist. In this paper we present a new view of matching vector codes and uncover certain similarities between MV codes and classical Reed-Muller (RM) codes. Our view allows us to obtain deeper insights into the power and limitations of MV codes. Specifically, we obtain the following: (1) We show that existing families of MV codes can be enhanced to tolerate a large constant fraction of errors, independent of the number of queries. Such enhancement comes at a price of a moderate increase in the number of queries. (2) Our construction yields the first families of MV codes of superconstant query complexity that can tolerate a constant fraction of errors. Our codes are shorter than RM LDCs for all values of $r\leq\log k/(\log\log k)^c$, for some constant $c$. (3) We show that any MV code encodes messages of length $k$ to codewords of length at least $k2^{\Omega(\sqrt{\log k})}$. Therefore MV codes do not improve upon RM LDCs for $r\geq(\log k)^{\Omega(\sqrt{\log k})}$.