Some results on the covering radii of Reed-Muller codes

Let R(r,m) be the rth-order Reed-Muller code of length 2^m and let ρ(r,m ) be its covering radius. R(2,7), R(2,8), R (3,7), and R(4,8) are among those smallest Reed-Muller codes whose covering radii are not known. New bounds for the covering radii of these four codes are obtained. The results are ρ(2,7)⩾40, ρ(2,8)⩾84, 20⩽ρ(3,7)⩽23, and ρ(4,8)⩾22. Noncomputer proofs for the known results that ρ(2,6)=18 and that R(1,5) is normal are given     

On the covering radius of codes

The covering radiusRof a code is the maximal distance of any vector from the code. This work gives a number of new results concerningt[n, k], the minimal covering radius of any binary code of lengthnand dimensionk. For examplet[n, 4]andt[n, 5]are determined exactly, and reasonably tight bounds ont[n, k]are obtained for anykwhennis large. These results are found by using several new constructions for codes with small covering radius. One construction, the amalgamated direct sum, involves a quantity called the norm of a code. Codes with normleq 2 R + 1are called normal, and may be combined efficiently. The paper concludes with a table giving bounds ont[n, k]forn leq 64.     

Divisibility properties for covering radius of certain cyclic codes

We are presenting a new method to obtain the covering radius of codes and in particular to prove quasi-perfection in codes. Our techniques combine divisibility results of Ax-Katz and Moreno-Moreno as well as coding theoretic methods. We answer a problem posed by Cohen-Honkala-Litsyn-Lobstein in the book covering radius for Bose-Chaudhuri-Hocquenghem (BCH) codes. We also obtain the covering radius for many new classes of codes.     

Short codes with a given covering radius

The covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword. The authors introduce a new function l(m,r), called the length function, which equals the smallest length of a binary code of codimension m and covering radius r. They investigate basic properties of the length function. Projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l(m,2) obtained by amalgamation of Hamming codes. General methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l(m,2). A table is presented which gives the best results now known for l(m,r) with m⩽12 and r⩽12     

Bounds for abnormal binary codes with covering radius one

The normality of binary codes is studied. The minimum cardinality of a binary code of length n with covering radius R is denoted by K(n,R). It is assumed that C is an (n,M)R code, that is, a binary code of length n with M codewords and covering radius R. It is shown that if C is an (n,M)1 code, then it is easy to find a normal (n ,M)1 code by changing C in a suitable way, and that all the optimal (n,M)1 codes (i.e. those for which M=K(n,1)) are normal and their every coordinate is acceptable. It is shown that if C is an abnormal (n,M) code, then n⩾9, and an abnormal (9118)1 code which is the smallest abnormal code known at present, is constructed. Lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived, and it is shown that if an (n,M)1 code is abnormal, then M⩾96     

Further results on the covering radius of codes

A number of upper and lower bounds are obtained forK(n, R), the minimal number of codewords in any binary code of lengthnand covering radiusR. Several new constructions are used to derive the upper bounds, including an amalgamated direct sum construction for nonlinear codes. This construction works best when applied to normal codes, and we give some new and stronger conditions which imply that a linear code is normal. An upper bound is given for the density of a covering code over any alphabet, and it is shown thatK(n + 2, R + 1) leq K(n, R)holds for sufficiently largen.     

New lower bounds for covering codes

Some new lower bounds on |C| for a binary linear [n, k]R code C with n+1=t(R +1)-r(0⩽r<R+1, t>2 odd) or with n+1=t(R+1)-1(t>2 even) are obtained. These bounds improve the sphere covering bound considerably and give several new values and lower bounds for the function t[n, k], the smallest covering radius of any [n, k] code     

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     

Improved sphere bounds on the covering radius of codes

The sphere bound is a trivial lower bound on K(n,R), the minimal cardinality of any binary code of length n and with covering radius R. By simple arguments it is considerably improved, to K(n,1)⩾2 ⁿ/n for n even. A table of lower and upper bounds on K(n,R) for n⩽33, R ⩽10 is included     

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     

On the covering radius of binary codes (Corresp.)

Upper bounds on the covering radius of binary codes are studied. In particular it is shown that the covering radiusr_{m}of the first-order Reed-Muller code of lenglh2^{m}satisfies2^{m-l}-2^{lceil m/2 rceil -1} r_{m} leq 2^{m-1}-2^{m/2-1}.     

Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem:A_i, the number of codewords of weightiin the second-order binary Reed-Muller code of length2^mis given byA_i = 0unlessi = 2^{m-1}or2^{m-1} pm 2^{m-l-j}, for somej, 0 leq j leq [m/2], A_0 = A_{2^m} = 1, and begin{equation} begin{split} A_{2^{m-1} pm 2^{m-1-j}} = 2^{j(j+1)} &{frac{(2^m - 1) (2^{m-1} - 1 )}{4-1} } \ .&{frac{(2^{m-2} - 1)(2^{m-3} -1)}{4^2 - 1} } cdots \ .&{frac{(2^{m-2j+2} -1)(2^{m-2j+1} -1)}{4^j -1} } , \ & 1 leq j leq [m/2] \ end{split} end{equation} begin{equation} A_{2^{m-1}} = 2 { 2^{m(m+1)/2} - sum_{j=0}^{[m/2]} A_{2^{m-1} - 2^{m-1-j}} }. end{equation}     

The second order Reed-Muller code of length 64 has covering radius 18 (Corresp.)

The assertion of the title is proved by first showing that the covering radius ofR(2, 6)cannot exceed18and then constructing a weight18coset leader.     

New generalizations of the Reed-Muller codes--I: Primitive codes

First it is shown that all binary Reed-Muller codes with one digit dropped can be made cyclic by rearranging the digits. Then a natural generalization to the nonbinary case is presented, which also includes the Reed-Muller codes and Reed-Solomon codes as special cases. The generator polynomial is characterized and the minimum weight is established. Finally, some results on weight distribution are given.     

The covering radius of the (128,8) Reed-Muller code is 56 (Corresp.)

Letr_{i}be the covering radius of the(2^{i},i+ 1)Reed-Muller code. It is an open question whetherr_{2m+1}=2^{2_{m}}-2mholds for allm. It is known to be true form=0,1,2, and here it is shown to be also true form=3.     

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     

Concatenated Reed-Muller codes for unequal error protection

We present a new coding scheme that combines the advantages of a product-like concatenation of Reed-Muller codes with so-called iterative “turbo” decoding and provides powerful unequal error protection abilities. It is shown that various levels of error protection can be realized using a sophisticated encoding scheme for Reed-Muller codes. A discussion of this code construction, the resulting distance profile between the different levels and the iterative decoding scheme is given. The results are very promising and impressively confirm the unequal error protection capabilities of the presented coding scheme     

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   

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     

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