Projective Reed-Muller codes

A class of codes in the Reed-Muller family, the projective Reed-Muller codes (PRM codes), is studied. The author defines the PRM codes of all orders and discusses their relation to polynomial codes. The exact parameters of PRM codes are given. The duals are characterized, and, in parallel to the classical works on generalized Reed-Muller codes, the cyclic properties are studied. Tables over parameters of the codes are given     

Weighted Reed-Muller codes and algebraic-geometric codes

A generalization of the Reed-Muller codes, the weighted Reed-Muller codes, is presented. The code parameters are estimated and the duals are shown also to be weighted Reed-Muller codes. It is shown how the minimum distance of certain algebraic-geometric codes in many cases can be determined exactly or an upper bound can be found, using subcodes which are weighted Reed-Muller codes     

Efficient decoding algorithms for generalized Reed-Muller codes

Previously, a class of generalized Reed-Muller (RM) codes has been suggested for use in orthogonal frequency-division multiplexing. These codes offer error correcting capability combined with substantially reduced peak-to mean power ratios. A number of approaches to decoding these codes have already been developed. Here, we present low complexity, suboptimal alternatives which are inspired by the classical Reed decoding algorithm for binary RM codes. We simulate these new algorithms along with the existing decoding algorithms using additive white Gaussian noise and two-path fading models for a particular choice of code. The simulations show that one of our new algorithms outperforms all existing suboptimal algorithms and offers performance that is within 0.5 dB of maximum-likelihood decoding, yet has complexity comparable to or lower than existing decoding approaches     

An average weight-distance enumerator for binary expansions ofReed-Solomon codes

An average Hamming weight enumerator is derived for the codewords at each Hamming distance from a received pattern in the set of all possible binary expansions of a Reed-Solomon code. Since these codes may be decoded by list decoders, such as those studied by Sudan (1997), the enumerator can be used to estimate the average number of codewords in the list returned by such a decoder     

Binary multilevel coset codes based on Reed-Muller codes

We study the construction and decoding of binary multilevel coset codes. This construction, originally introduced by Blokh and Zyablov in 1974 and by Zinov'ev in 1976, shows remarkable analogies with most recent schemes of coded modulations. Basic elements of the construction are an inner code, head of a partition chain having suitable distance properties, and a set of outer codes, generally nonbinary. For each partition level there is an outer code whose alphabet has the same order of the partition: in this way it is possible to associate every partition subset to a code symbol. It is well known that these codes can be efficiently decoded by the so called “multistage decoding.” We show that good codes (in terms of performance/complexity) can be constructed using Reed-Muller (RM) codes as inner codes. To this aim RM codes are revisited in the framework of the above construction and decoding techniques. In particular we describe a family of decoders for RM codes which include Forney's (1988) and Hemmati's (1989) decoders as special cases. Finally, we present some examples of efficient binary codes based on RM codes, and assess their performance via computer simulation     

Trellis based code combining protocols for Reed-Muller codes

A novel code combining system based on Reed-Muller codes is presented. Because of their simple structure RM codes are simple to decode using a trellis based soft maximum likelihood decoder (SMLD). The decoder exploits the modular structure of the RM code to construct a set of nested trellises which minimise the complexity of the decoder by re-using the results of previous decoding attempts. A protocol utilising this technique to produce an efficient code combining ARQ-scheme is also introduced     

Soft-decision decoding of Reed-Muller codes as generalized multipleconcatenated codes

Constructs Reed-Muller codes by generalized multiple concatenation of binary block codes of length 2. As a consequence of this construction, a new decoding procedure is derived that uses soft-decision information. The algorithm is designed for low decoding complexity and is applicable to all Reed-Muller codes. It gives better decoding performance than soft-decision bounded-distance decoding. Its decoding complexity is much lower than that of maximum-likelihood trellis decoding of Reed-Muller codes, especially for long codes     

Soft-decision majority decoding of Reed-Muller codes

We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2^m transmitted symbols are recalculated into the reliabilities of 2^m-r parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log₁₀(π/2)≈2 dB at any given output error probability. For fixed code rate R and m→∞, our algorithm increases almost 2^r/2 times the correcting capability of soft-decision bounded distance decoding     

List decoding of q-ary Reed-Muller codes

The q-ary Reed-Muller (RM) codes RM/sub q/(u,m) of length n=q/sup m/ are a generalization of Reed-Solomon (RS) codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for RM codes were given in and . The algorithm in Sudan et al. (1999) is an improvement of the algorithm in , it is applicable to codes RM/sub q/(u,m) with u相似文献   

New covering radius of Reed-Muller codes for t-resilient functions

In this paper, we introduce a new covering radius of RM(r,n) from cryptography viewpoint. It is defined as the maximum distance between t-resilient functions and the rth order Reed-Muller code RM(r,n). We next derive its lower and upper bounds. We further present a table of numerical data of our bounds.     

Recursive decoding and its performance for low-rate Reed-Muller codes

Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length n and fixed order r. An algorithm is designed that has complexity of order nlogn and corrects most error patterns of weight up to n(1/2-/spl epsiv/) given that /spl epsiv/ exceeds n/sup -1/2r/. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity. To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length n grows.     

Generalized Hamming weights of q-ary Reed-Muller codes

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as well as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition     

On the weight structure of Reed-Muller codes

The following theorem is proved. Letf(x_1,cdots, x_m)be a binary nonzero polynomial ofmvariables of degreenu. H the number of binarym-tuples(a_1,cdots, a_m)withf(a_1, cdots, a_m)= 1 is less than2^{m-nu+1}, thenfcan be reduced by an invertible affme transformation of its variables to one of the following forms. begin{equation} f = y_1 cdots y_{nu - mu} (y_{nu-mu+1} cdots y_{nu} + y_{nu+1} cdots y_{nu+mu}), end{equation} wherem geq nu+muandnu geq mu geq 3. begin{equation} f = y_1 cdots y_{nu-2}(y_{nu-1} y_{nu} + y_{nu+1} y_{nu+2} + cdots + y_{nu+2mu -3} y_{nu+2mu-2}), end{equation} This theorem completely characterizes the codewords of thenuth-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results.     

Performance of Reed-Muller codes and a maximum-likelihood decodingalgorithm for OFDM

The performance of Reed-Muller encoding and a maximum-likelihood decoding algorithm for orthogonal frequency-division multiplexing is presented. The example codes have a tightly bounded peak-to-mean envelope power ratio, while simultaneously enabling powerful error correction. We present a maximum-likelihood decoder that makes use of a distance-preserving map and multiple fast Hadamard transforms. Its operation is described in detail and its performance is assessed under realistic channel conditions     

Orphans of the first order Reed-Muller codes

If C is a code, an orphan is a coset that is not a descendant. Orphans arise naturally in the investigation of the covering radius. Case C has only even-weight vectors and minimum distance of at least four. Cosets that are orphans are characterized, and then the existence is proved of a family of orphans of first-order Reed-Muller codes R(1, m). For m⩽5 all orphans of R(1, m) are identified     

Augmented product codes and lattices: Reed-Muller codes and Barnes-Wall lattices

This paper concerns the construction of the so-called augmented product codes and augmented product lattices. These are obtained by augmenting product codes or product lattices from certain classes thus obtaining higher dimensional codes or lattices from the same class, respectively. Certain properties of the augmented product construction are derived, and specific construction examples are given. In particular, it is shown that the Reed-Muller codes, the Golay code, the Barnes-Wall lattices, as well as the Leech lattice all have various augmented product constructions.     

More results on the weight enumerator of product codes

We consider the product code C/sub p/ of q-ary linear codes with minimum distances d/sub c/ and d/sub r/. The words in C/sub p/ of weight less than d/sub r/d/sub c/+max(d/sub r//spl lceil/(d/sub c//g)/spl rceil/,d/sub c//spl lceil/(d/sub r//q)/spl rceil/) are characterized, and their number is expressed in the number of low-weight words of the constituent codes. For binary product codes, we give an upper bound on the number of words in C/sub p/ of weightless than min(d/sub r/(d/sub c/+/spl lceil/(d/sub c//2)/spl rceil/+1)), d/sub c/(d/sub r/+/spl lceil/(d/sub r//2)/spl rceil/+1) that is met with equality if C/sub c/ and C/sub r/ are (extended) perfect codes.     

Simple maximum-likelihood decoding of generalized first-order Reed-Muller codes

An efficient decoder for the generalized first-order Reed-Muller code RM/sub q/ (1, m) is essential for the decoding of various block-coding schemes for orthogonal frequency-division multiplexing with reduced peak-to-mean power ratio. We present an efficient and simple maximum-likelihood decoding algorithm for RM/sup q/ (1, m). It is shown that this algorithm has lower complexity than other previously known maximum-likelihood decoders for RM/sub q/ (1, m).     

Automorphism groups of homogeneous and projective Reed-Muller codes

We characterize the full automorphism groups of homogeneous and projective Reed-Muller (HRM and PRM) codes. These groups are respectively related to general and projective linear groups     

A novel turbo coding scheme for satellite ATM using Reed-Muller codes

Block turbo codes with trellis-based decoding are proposed for use in cell-based satellite communication. Shortened Reed-Muller (RM) codes are used as the component codes because their minimal trellis is known. Simulation results for RM turbo codes and shortened RM turbo codes are presented over additive white Gaussian noise and Rayleigh fading channels. The performance of the shortened codes with different shortening patterns are shown. In some cases, the codes have the unequal error protection property, useful in asynchronous transfer mode cell formatting. In order to test the suitability of the proposed coding scheme from a practical point of view, the effect of channel impairments, including channel signal-to-noise ratio mismatch and carrier phase offset, are investigated.