Mubarak Jibril

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.

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.

A Generalized Construction of Extended Goppa Codes

We present a generalized construction of extended length Goppa codes. Using this construction, we obtain 71 new codes in finite field Fq for q = 4, 7, 8, 9 with better minimum distance than the previously known codes with the same length and dimension.

Some new codes from binary Goppa codes and a method of shortening linear codes

Goppa codes have some of the largest minimum distances possible for linear codes. The authors use some binary Goppa codes in which four new binary codes are obtained with parameters better than any codes known to date. The authors also present the necessary conditions for a code obtained by shortening a longer code which will have a greater minimum distance than the original code.

Performance comparison between Hermitian codes and shortened non-binary BCH codes

We explore the benefits of implementing Hermitian codes over moderate lengths and compare the hard decision decoding performance in the additive white Gaussian noise (AWGN) channel and performance in the erasure channel of Hermitian codes with shortened non-binary Bose Hocquenghem Chaudhuri (BCH) codes. We implement the Berlekamp-Massey-Sakata (BMSA) decoding and Berlekamp-Massey (BMA) decoding for the hard decision Hermitian and BCH codes respectively, maximum likelihood erasure decoding and ordered reliability soft decision decoding for both.

Good Binary Linear Codes

Two of the important performance indicators for a linear code are the minimum Hamming distance and the weight distribution. This chapter is concerned with methods and tools for producing new codes which are improvements to currently known codes. The chapter discusses and describes several efficient algorithms for computing the minimum distance and weight distribution of linear codes. Several different methods of constructing codes are described and several new codes are presented which are improvements to the codes listed in the database of best-known codes. Methods of combining known codes to produce good codes are also explored in detail. These methods are applied using cyclic codes as components to the new codes and several new binary codes are found as a result.

Combined Error Detection and Error-Correction

This chapter is concerned with the design of codes and systems for combined error detection and correction, primarily aimed at applications featuring retransmission of data packets which have not been decoded correctly. Several such hybrid automatic request, HARQ, systems are described including a novel system variation which uses a retransmission metric based on a soft decision; the Euclidean distance between the decoded codeword and the received vector. It is shown that a cyclic redundancy check, CRC, is not essential for this system and need not be transmitted. The generator matrix of a nested set of block codes of length 251 bits is constructed by applying construction X three times in succession starting with an extended BCH (128, 113, 6) code. The result is the basis for an incremental-redundancy system whereby the first 128 bits transmitted is a codeword from the BCH code, followed by the transmission of a further 32 bits, if requested, producing a codeword from a (160, 113, 12) code. Further requests finally result in a codeword from a (251, 113, 20) code. Each request increases the chance of correct decoding by increasing the minimum Hamming distance of the net received codeword. Performance graphs are presented showing the comparative error rate performances and throughputs of the new systems compared to the standard HARQ system. Lower error floors and increased throughputs are evident.

Algebraic Geometry Codes

This chapter introduces algebraic geometry (AG) codes. Algebraic geometry codes have good properties and some families of these codes have been shown to be asymptotically good. Concepts and notions relevant to AG codes are discussed with particular emphasis laid on the construction of AG codes. A generalised construction of AG codes is introduced and improvements to the tables of best-known codes are presented. This construction utilises the notion of places of higher degree of a curve as well as a new concatenation concept.

Reed–Solomon Codes and Binary Transmission

This chapter further discusses the Reed–Solomon codes which are exceptional, symbol based, because they are maximum distance separable (MDS) codes. Though these codes are not binary codes, they can be used as binary codes. The performance of Reed–Solomon codes when used on a binary channel is explored in this chapter and compared to codes which are designed for binary transmission. The construction of the parity-check matrices of RS codes for use as binary codes is described in detail for specific code examples. The simulation results for three differently constructed (150, 75) codes for the binary AWGN channel using a near maximum likelihood decoder are presented. Surprisingly the best performing code at \(10^{-4}\) error rate is not the best-known (150, 75, 23) code. It is shown that this is due to the higher multiplicities of weight 32–36 codeword errors. However, beyond \(10^{-6}\) error rates the best-known (150, 75, 23) code is the best code.