On a class of majority-logic decodable cyclic codes

A new infinite class of cyclic codes is studied. Codes of this class can be decoded in a step-by-step manner, using` majority logic. Some previously known codes fall in this class, and thus admit simpler decoding procedures. As random error-correcting codes, the codes are nearly as powerful as the Bose-Chaudhuri codes.     

On binary majority-logic decodable codes (Corresp.)

LetVprimebe a binary(n,k)majority-logic decodable code withgprime (X)as its generator polynomial and odd minimum distanced. LetVbe the(n, k - 1)subset code generated bygprime (X)(1 + X). This correspondence shows thatVis majority-logic deeodable withd + 1orthogonal estimates. This fact is useful in the simultaneous correction of random errors and erasures.     

New family of binary one-step majority-logic decodable codes

One-step completely orthogonalisable binary codes derived from block designs are presented and their decoder structure is given.<>     

一种适用于大数逻辑可译LDPC码的自适应译码算法

大数逻辑可译低密度奇偶校验(LDPC)码是一类具有较大列重的码,针对此类特殊的LDPC码,提出了一种基于整数可靠度的低复杂度自适应译码算法。在译码的过程中,算法对每个校验节点分别引入不同的自适应修正因子对外信息进行修正。仿真表明提出的自适应译码算法的性能与和积译码算法的性能相当,在误码率(BER)约为10-5时两种算法性能之间仅有0.1 d B的差异。所提算法具有复杂度低、可并行操作、全整数的信息传递等优点,十分有利于工程实现。     

A note on one-step majority-logic decodable codes (Corresp.)

Construction of shortened geometric codes as Shown here results in 1-step majority-logic deeodable codes. The shortened codes retain the error-correction ability of the parent codes and the decoders for the shortened codes are much simpler than for the parent code. A table of shortened codes is given.     

一类一步多数逻辑可译码的广义汉明重量谱

线性码的广义汉明重量谱描述了码在第二类窃密信道中传输的密码学特征。该文针对一类循环码在仿射置换群之下不变的一步多数逻辑可译码的广义汉明重量谱进行了研究,提出了该类码的重量谱的估计方法,并通过实例作了说明。     

On efficient majority logic decodable codes

A particular shortening technique is applied to majority logic decodable codes of length2^{t}. The shortening technique yields new efficient codes of lengthsn = 2^{p}, wherepis a prime, e.g., a (128,70) code withd_{maj} = 16. For moderately long code lengths (e.g.,n = 2^{11} or 2^{13}), a 20-25 percent increase in efficiency can be achieved over the best previously known majority logic decodable codes. The new technique also yields some efficient codes for lengthsn = 2^{m}wheremis a composite number, for example, a (512,316) code withd_{maj} = 32code which has 42 more information bits than the previously most efficient majority logic decodable code.     

New construction of majority logic decodable quasicyclic codes

A new construction of majority logic decodable quasicyclic codes is presented. As an example, an infinite family of quasicyclic codes with a minimum distance of four is constructed, and comparisons are made to show that they are better than selforthogonal quasicyclic codes and Shiva codes.<>     

On majority-logic decoding of finite geometry codes

In this paper, an improved decoding algorithm for codes that are constructed from finite geometries is introduced. The application of this decoding algorithm to Euclidean geometry (EG) and projective geometry (PG) codes is further discussed. It is shown that these codes can be orthogonalized in less than or equal to three steps. Thus, these codes are majority-logic decodable in no more than three steps. Our results greatly reduce the decoding complexity of EG and PG codes in most cases. They should make these codes very attractive for practical use in error-control systems.     

A class of high-speed decodable burst-correcting codes (Corresp.)

A class of high-speed decodable burst-correcting codes is presented. This class of codes is obtained by modifying burst-correcting convolutional codes into block codes and does not require any cyclic shifts in the decoding process. With the appropriate choices of parameters, the codes can approximate minimum-redundancy codes. The high-speed decodability is expected to make these codes suitable for application to computer systems.     

On majority-logic decoding for duals of primitive polynomial codes

The class of polynomial codes introduced by Kasami et al. has considerable inherent algebraic and geometric structure. It has been shown that this class of codes and their dual codes contain many important classes of cyclic codes as subclasses, such as BCH codes, Reed-Solomon codes, generalized Reed-Muller codes, projective geometry codes, and Euclidean geometry codes. The purpose of this paper is to investigate further properties of polynomial codes and their duals. First, majority-logic decoding for the duals of certain primitive polynomial codes is considered. Two methods of forming nonorthogonal parity-check sums are presented. Second, the maximality of Euclidean geometry codes is proved. The roots of the generator polynomial of an Euclidean geometry code are specified.     

Iteratively maximum likelihood decodable spherical codes and amethod for their construction

The authors propose a class of spherical codes which can be easily decoded by an efficient iterative maximum likelihood decoding algorithm. A necessary and sufficient condition for a spherical code to be iteratively maximum likelihood decodable is formulated. A systematic construction method for such codes based on shrinking of Voronoi corners is analyzed. The base code used for construction is the binary maximal length sequence code. The second-level construction is described. Computer simulation results for selected codes constructed by the proposed method are given     

Easily decodable binary cyclic codes

This paper presents a class of binary cyclic codes with block length n = 2^m? 1, having n?k = 2^m? parity checks and a minimum distance d=m +1, where m is an integer. These codes are shown to be majority logic decodable in one step by making use of the concept of a quasi-perfect finite difference set.     

Easily decodable efficient self-orthogonal block codes

A class of self-orthogonal block codes is described, which are easy to decode and to augments. They are constructed by forming parity-check equations across certain patterns of information digits arranged in a 2-dimensional array. The codes are of a rate close to the optimum for self-othogonal codes.     

Class of majority decodable real-number codes

A majority decoding algorithm for a class of real-number codes is presented. Majority decoding has been a relatively simple and fast decoding technique for codes over finite fields. When applied to decode real-number codes, the robustness of the majority decoding to the presence of background noise, which is usually an annoying problem for existing decoding algorithms for real-number codes, is its most prominent property. The presented class of real-number codes has generator matrices similar to those of the binary Reed-Muller codes and is decoded by similar majority logic     

On the Lin-Weldon majority-logic decoding algorithm for product codes (Corresp.)

It is shown that a majority-logic decoding algorithm proposed by Lin and Weldon for the product of anL-step and a one-step orthogonalizable code is incomplete whenLis greater than unity. An improvement is presented to overcome this disadvantage in the binary case.     

Algorithms for majority-logic check equations of maximal-length codes

Further properties of maximal-length codes are briefly descnbed. These are shown to be of value in deriving the majority-logic check equations. Examples illustrating the effectiveness of the algorithms are given for (15, 4) and (31, 5) maximal-length codes.     

Uniquely decodable codes for deterministic relay channels

The deterministic relay channel is analyzed and explicit code constructions for all binary and all ternary/binary channels are given. An explicit set of equivalence conditions is used to make a classification of all such relay channels, for which also the capacity is evaluated. The coding problem is then reduced to finding all possible output sequences of a certain finite-state channel determined by the relay coding strategy. The channel states correspond to the possible relay memory contents. For some relay channels capacity is reached by using simple uniquely decodable codes, thus establishing the zero-error capacity of those channels with finite-memory relay strategies. For other relay channels the relay memory must be arbitrarily large to achieve zero-error rates arbitrarily close to capacity. One such code construction is given. It is not known whether there exist relay channels for which the zero-error capacity is strictly smaller than the average-error capacity. The code construction problem for the semideterministic relay channel and for the nonsynchronized relay channel is briefly considered     

Distributed decoding of cyclic block codes using a generalizationof majority-logic decoding

One-step majority-logic decoding is one of the simplest algorithms for decoding cyclic block codes. However, it is an effective decoding scheme for very few codes. This paper presents a generalization based on the “common-symbol decoding problem.” Suppose one is given M (possibly corrupted) codewords from M (possibly different) codes over the same field; suppose further that the codewords share a single symbol in common. The common-symbol decoding problem is that of estimating the symbol in the common position. This is equivalent to one-step majority logic decoding when each of the “constituent” codes is a simple parity check. This paper formulates conditions under which this decoding is possible and presents a simple algorithm that accomplishes the same. When applied to decoding cyclic block codes, this technique yields a decoder structure ideal for parallel implementation. Furthermore, this approach frequently results in a decoder capable of correcting more errors than one-step majority-logic decoding. To demonstrate the simplicity of the resulting decoders, an example is presented     

Linear-time encodable/decodable codes with near-optimal rate

We present an explicit construction of linear-time encodable and decodable codes of rate r which can correct a fraction (1-r-/spl epsiv/)/2 of errors over an alphabet of constant size depending only on /spl epsiv/, for every 00. The error-correction performance of these codes is optimal as seen by the Singleton bound (these are "near-MDS" codes). Such near-MDS linear-time codes were known for the decoding from erasures; our construction generalizes this to handle errors as well. Concatenating these codes with good, constant-sized binary codes gives a construction of linear-time binary codes which meet the Zyablov bound, and also the more general Blokh-Zyablov bound (by resorting to multilevel concatenation). Our work also yields linear-time encodable/decodable codes which match Forney's error exponent for concatenated codes for communication over the binary symmetric channel. The encoding/decoding complexity was quadratic in Forney's result, and Forney's bound has remained the best constructive error exponent for almost 40 years now. In summary, our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding, but in addition have linear-time encoding and decoding algorithms.