On the construction of a class of majority-logic decodable codes

The attractiveness of majority-logic decoding is its simple implementation. Several classes of majority-logic decodable block codes have been discovered for the past two decades. In this paper, a method of constructing a new class of majority-logic decodable block codes is presented. Each code in this class is formed by combining majority-logic decodable codes of shorter lengths. A procedure for orthogonalizing codes of this class is formulated. For each code, a lower bound on the number of correctable errors with majority-logic decoding is obtained. An upper bound on the number of orthogonalization steps for decoding each code is derived. Several majority-logic decodable codes that have more information digits than the Reed-Muller codes of the same length and the same minimum distance are found. Some results presented in this paper are extensions of the results of Lin and Weldon [11] and Gore [12] on the majority-logic decoding of direct product codes.     

Multifold Euclidean geometry codes

This paper presents a class of majority-logic decodable codes whose structure is based on the structural properties of Euclidean geometries (EG) and codes that are invariant under the affine group of permutations. This new class of codes contains the ordinary EG codes and some generalized EG codes as subclasses. One subclass of new codes is particularly interesting: they are the most efficient majority-logic decodable codes that have been constructed.     

Polynomial codes

A class of cyclic codes is introduced by a polynomial approach that is an extension of the Mattson-Solomon method and of the Muller method. This class of codes contains several important classes of codes as subclasses, namely, BCH codes, Reed-Solomon codes, generalized primitive Reed-Muller codes, and finite geometry codes. Certain fundamental properties of this class of codes are derived. Some subclasses are shown to be majority-logic decodable.     

Shortened finite geometry codes (Corresp.)

A method of shortening finite analytic geometry codes, projective-geometry (PG) codes, Euclidean-geometry (EG) codes, and 2-fold EG codes is presented. The shortened codes preserve the feature of being majority-logic decodable and they have the same error-correcting capability as the original codes. Combinatorial expressions for the parity-check symbols of the shortened codes are derived.     

On the stopping distance of finite geometry LDPC codes

In this letter, the stopping sets and stopping distance of finite geometry LDPC (FG-LDPC) codes are studied. It is known that FG-LDPC codes are majority-logic decodable and a lower bound on the minimum distance can be thus obtained. It is shown in this letter that this lower bound on the minimum distance of FG-LDPC codes is also a lower bound on the stopping distance of FG-LDPC codes, which implies that FG-LDPC codes have considerably large stopping distance. This may explain in one respect why some FG-LDPC codes perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs.     

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.     

Error-erasure decoding of product codes (Corresp.)

Two error-erasure decoding algorithms for product codes that correct all the error-erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given. The first algorithm works when at least one of the component codes is majority-logic decodable. The second algorithm works for any product code. Both algorithms use the decoders of the component codes.     

Generalized tensor product codes

A class of binary error-correcting codes, called generalized tensor product codes, is presented with their decoding algorithm. These codes are constructed by combining a number of codes on various extension fields with shorter binary codes. A general algorithm is provided to do bounded distance decoding for these codes. Simply decodable codes such as Wolf's tensor product codes are shown to be special cases of this class of codes. Simply decodable and more efficient codes than Wolf's codes are also included as special cases.     

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.     

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。     

Errors-and-erasures decoding of binary majority-logic-decodable codes

The decoder of a binary majority-logic decodable code can be modified to enable the correction of erasures as well as errors. The changes required in both type-I and type-II decoders are described and compared in terms of increases in decoder complexity and internal clock rates.     

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.     

Single-symbol maximum likelihood decodable linear STBCs

Space-time block codes (STBCs) from orthogonal designs (ODs) and coordinate interleaved orthogonal designs (CIOD) have been attracting wider attention due to their amenability for fast (single-symbol) maximum-likelihood (ML) decoding, and full-rate with full-rank over quasi-static fading channels. However, these codes are instances of single-symbol decodable codes and it is natural to ask, if there exist codes other than STBCs form ODs and CIODs that allow single-symbol decoding? In this paper, the above question is answered in the affirmative by characterizing all linear STBCs, that allow single-symbol ML decoding (not necessarily full-diversity) over quasi-static fading channels-calling them single-symbol decodable designs (SDD). The class SDD includes ODs and CIODs as proper subclasses. Further, among the SDD, a class of those that offer full-diversity, called Full-rank SDD (FSDD) are characterized and classified. We then concentrate on square designs and derive the maximal rate for square FSDDs using a constructional proof. It follows that 1) except for N=2, square complex ODs are not maximal rate and 2) a rate one square FSDD exist only for two and four transmit antennas. For nonsquare designs, generalized coordinate-interleaved orthogonal designs (a superset of CIODs) are presented and analyzed. Finally, for rapid-fading channels an equivalent matrix channel representation is developed, which allows the results of quasi-static fading channels to be applied to rapid-fading channels. Using this representation we show that for rapid-fading channels the rate of single-symbol decodable STBCs are independent of the number of transmit antennas and inversely proportional to the block-length of the code. Significantly, the CIOD for two transmit antennas is the only STBC that is single-symbol decodable over both quasi-static and rapid-fading channels.     

Ternary graph theoretic error-correcting codes (Corresp.)

It is shown that directed graphs can be used to generate a class of moderately efficient error-correcting codes, which are easily decodable.     

Improved Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: i) the codes lie close to the Singleton bound, ii) they can be encoded in time complexity that is linear in their code length, and iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace maximum-distance separable (MDS) outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size     

Spectrum of incorrectly decoded bursts for cyclic burst error codes

The enumeration of the incorrectly decoded bursts for cyclic burst error-correcting codes is reported here. The enumeration yields closed formulas for long bursts, whereas an efficient algorithm for the enumeration is given for the short bursts. The enumeration is carried out for two different decoding rules. Under the first rule, the cyclic code is a full length code, and split decodable patterns are decoded by the decoder in spite of the fact that a split decodable pattern can be a burst that exceeds the decoding capability of the decoder. The analysis for the second rule assumes split patterns are not decoded. This analysis is valid for the class of shortened cyclic codes.     

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     

Application of circulant matrices to the construction and decodingof linear codes

The Fourier transform technique is used to analyze and construct several families of double-circulant codes. The minimum distance of the resulting codes is lower-bounded by 2√r and can be decoded easily employing the standard BCH decoding algorithm or the majority-logic decoder of Reed-Muller codes. A decoding procedure for Reed-Solomon codes is presented, based on a representation of the parity-check matrix by circulant blocks. The decoding procedure inherits both the (relatively low) time complexity of the Berlekamp-Massey algorithm and the hardware simplicity characteristic of Blahut's algorithm. The procedure makes use of the encoding circuit together with a reduced version of Blahut's decoder     

Development of an Error-Correction Method for Data Packet Multiplexed with TV Signals

By using the vertical blanking period of television signals, it is possible to transmit coded data such as teletext, telesoftware, music, etc. However, the quality of data transmission on television transmission channels is very poor and a powerful error-correction code is required to reliably transmit coded data. From the results of simulations using error pattern data collected in field tests and the comparison of various error-correction codes under many conditions, it has been determined that the shortened (272, 190) majority-logic decodable difference-set cyclic code is a suitable code for NTSC TV signals. By using error-correction codes proposed to date for teletext, it has been difficult to obtain a page error rate (PER) of 10^-1in many measurement points. However, PER's of less than 10^-2can be obtained in this system, even when random noise, ghost interference, or waveform distortion are present and bit error rates (BER's) are 10^-2. This paper also gives an empirical equation according to the error data and shows that the error-correction capability increased equivalently up to 11 error-bits in a packet by modifying the decoding algorithm.     

Braided Convolutional Codes: A New Class of Turbo-Like Codes

We present a new class of iteratively decodable turbo-like codes, called braided convolutional codes. Constructions and encoding procedures for tightly and sparsely braided convolutional codes are introduced. Sparsely braided codes exhibit good convergence behavior with iterative decoding, and a statistical analysis using Markov permutors shows that the free distance of these codes grows linearly with constraint length, i.e., they are asymptotically good.