Cen Jung Tjhai

Analysis of the distribution of the number of erasures correctable by a binary linear code and the link to low-weight codewords

The number and weight of low-weight codewords of a binary linear code determine the erasure channel performance. Analysis is given of the probability density function of the number of erasures correctable by the code in terms of the weight enumerator polynomial. For finite-length codes, zero erasure decoder error rate is impossible, even with maximum-distance-separable (MDS) codes and maximum-likelihood decoding. However, for codes that have binomial weight spectra, for example BCH, Goppa and double-circulant codes, the erasure correction performance is close to that of MDS codes. One surprising result is that, for many (n, k) codes, the average number of correctable erasures is almost equal to n-k, which is significantly larger than d min -1. For the class of iteratively decodable codes (LDPC and turbo codes), the erasure performance is poor in comparison to algebraic codes designed for maximum d min . It is also shown that the turbo codes that have optimised d min have significantly better performance than LDPC codes. A probabilistic method, which has considerably smaller search space than that of the generator matrix-based methods, is presented to determine the d min of a linear code using random erasure patterns. Using this approach, it is shown that there are (168, 84, 24) and (216, 108, 24) quadratic double-circulant codes

Improved Data Recovery from Patterned Media With Inherent Jitter Noise Using Low-Density Parity-Check Codes

Patterned magnetic media promises areal densities in excess of 1 Tbit/in2 for data storage. However, current imperfect patterning techniques result in a variation in the dimensions and distribution of the fabricated islands. As a result, this variation introduces jitter in the replay waveform that makes data recovery difficult. In this paper, we investigate the use of low-density parity-check (LDPC) codes and iterative decoding for mitigating the effects of lithography jitter and improving the read channel performance in patterned media storage systems. In addition, we show that the adoption of LDPC coding techniques permits an increase in the data storage capability of the medium to approximately 1.6 Tbit/in2 with acceptable bit-error-rate performance.

Some Results on the Weight Distributions of the Binary Double-Circulant Codes Based on Primes

This paper presents a more efficient algorithm to count codewords of given weights in self-dual double-circulant and formally self-dual quadratic double-circulant codes over GF(2). A method of deducing the modular congruence of the weight distributions of the binary quadratic double-circulant codes is proposed. This method is based on that proposed by Mykkeltveit, Lam and McEliece, JPL. Tech. Rep., 1972, which was applied to the extended quadratic-residue codes. A useful application of this modular congruence method is to provide independent verification of the weight distributions of the extended quadratic-residue and quadratic double-circulant codes. Using this method in conjunction with the proposed efficient codeword counting algorithm, we are able i) to give the previously unpublished weight distributions of the [76, 38,12] and [124, 62, 20] binary quadratic double-circulant codes; ii) to provide corrections to the published results on the weight distributions of the binary extended quadratic-residue code of prime 151, and the number of codewords of weights 30 and 32 of the binary extended quadratic-residue code of prime 137; and iii) to prove that the [168, 84, 24] extended quadratic-residue and quadratic double-circulant codes are inequivalent

Extending the Dorsch decoder towards achieving maximum-likelihood decoding for linear codes

It is shown that the relatively unknown Dorsch decoder may be extended to produce a decoder that is capable of maximum-likelihood decoding. The extension involves a technique for any linear (n, k) code that ensures that n−k less reliable, soft decisions of each received vector may be treated as erasures in determining candidate codewords. These codewords are derived from low information weight codewords and it is shown that an upper bound of this information weight may be calculated from each received vector in order to guarantee that the decoder will achieve maximum-likelihood decoding. Using the cross-correlation function, it is shown that the most likely codeword may be derived from a partial correlation function of these low information weight codewords, which leads to an efficient fast decoder. For a practical implementation, this decoder may be further simplified into a concatenation of a hard-decision decoder and a partial correlation decoder with insignificant performance degradation. Results are presented for some powerful, known codes, including a GF(4) non-binary BCH code. It is shown that maximum-likelihood decoding is realised for a high percentage of decoded codewords and that performance close to the sphere packing bound is attainable for codeword lengths up to 1000 bits.

A Generalized Construction and Improvements on Nonbinary Codes From Goppa Codes

We present an efficient construction of extended length Goppa codes. Using this construction, we obtain 78 new nonbinary codes with better minimum distance than the previously known codes with the same length and dimension. The construction is based on the observation that certain Goppa codes can be seen as BCH codes.

Good codes from generalised algebraic geometry codes

Algebraic geometry codes or Goppa codes are defined with places of degree one. In constructing generalised algebraic geometry codes places of higher degree are used. In this paper we present 41 new codes over F16 which improve on the best known codes of the same length and rate. The construction method uses places of small degree with a technique originally published over 10 years ago for the construction of generalised algebraic geometry codes.

Novel Soft-Feedback Equalization Method for Multilevel Magnetic Recording

This paper investigates the use of multilevel modulation for magnetic recording using a novel soft-feedback equalization (SFE) approach. Different aspects of investigation are 1)multilevel recording, 2) SFE, and 3) application of turbo codes. The SFE scheme is a model in which the partial response (PR) equalizer and maximum a posteriori (MAP) decoder are replaced by a linear filter with an iterative MAP decoder. Error correction codes (ECCs) are applied to the multilevel recording system in order to achieve very low error rates. Implementation of the SFE scheme for multilevel recording shows a reduction in complexity in comparison to various PRML schemes. The simulation results show a clear performance gain of multi-level-coded against binary-coded recording systems. At higher signal-to-noise ratio (SNR), the coded multilevel SFE scheme overcomes the error floor effect produced in the coded multilevel PRML scheme, which is caused by minimum distance error events. Overall, this paper proposes the use of coded multilevel recording with SFE scheme at lower rates rather than coded binary recording at higher densities in order to achieve similar performance

Comparison of Concatenated Reed-Solomon Coding with Hard-decision to Soft-decision LDPC Coding for the Rayleigh Fading Channel

Reed-Solomon (RS) codes C(n, k, n - k + 1) are commonly used error-control codes, because they are Maximum Distance Separable (MDS) codes. Over a Binary Erasure Channel, RS codes perform with optimal results and approach the maximum channel capacity. In this paper, we apply RS codes to packet wireless transmission over the uncorrelated flat Rayleigh fading channel. It is shown that by using an RS code concatenated with BCH codes and using hard decisions, better results are obtained than using bit interleaved LDPC codes, with soft-decision decoding. The BCH code is used to correct small numbers of errors due to noise and also to detect the presence of deep fades, in which case the entire packet is erased. Erased packets are corrected by the RS code. We also discuss the effect of overall code rate on the net performance.

On the Efficient Codewords Counting Algorithm and the Weight Distributions of the Binary Quadratic Double-Circulant Codes

An efficient algorithm to count all codewords of given weight and a method to deduce the modular congruence of the weight distributions of the binary quadratic double-circulant codes are presented. Using this algorithm, we give the weight distribution of the quadratic double-circulant code of length 168 and that of the extended quadratic-residue code of the same length. The weight distributions of these two inequivalent codes of length 168, which were previously unknown, are independently verified and proved to be accurate using the modular congruence method

New binary codes from extended Goppa codes

An efficient construction of extended length Goppa codes is presented. The construction yields four new binary codes [153, 71, 25], [151, 70, 25], [160, 70, 27], and [158, 69, 27]. The minimum distances are larger than those of the best previously known linear codes of the same length and dimension.