Binary cyclic codes from three classes of sequences

This paper describes how 2-adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over Z/sub 2(a)/, a/spl ges/2, the ring of integers modulo 2/sup a/. It provides a 2-adic proof of a theorem of McEliece that characterizes the possible Hamming weights that can appear in a binary cyclic code. A generalization of this theorem is derived that applies to cyclic codes over Z/sub 2(a)/ that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2/sup a/ appears as a component of an arbitrary codeword in the cyclic code. The limit of the sequence of Hensel lifts is a universal code defined over the 2-adic integers. This code was first introduced by Calderbank and Sloane (1995), and is the main subject of this paper. Binary cyclic codes and cyclic codes over Z/sub 2(a)/ are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over Z/sub 4/ that are obtained from binary cyclic codes by means of a single Hensel lift. The binary images of such codes under the Gray isometry include the Kerdock, Preparata, and Delsart-Goethals codes. These are nonlinear binary codes that contain more codewords than any linear code presently known. Fundamental understanding of the composition of codewords in cyclic codes over Z/sub 4/ is central to the search for more families of optimal codes. This paper also constructs even unimodular lattices from the Hensel lift of extended binary cyclic codes that are self-dual with all Hamming weights divisible by 4. The Leech lattice arises in this way as do extremal lattices in dimensions 32 through 48.

Projection of coded patterns is a popular method for the acquisition of dense range images. Usually, the binary code or gray code is used for projection patterns. In this paper we investigate the light plane identification error in the two codes. Our main result is that the binary code and gray code have the same error characteristic while the partial gray code is superior to the partial binary code.

The sensitivity of a binary block code to loss of synchronism (misplacement of the "commas" separating codewords) can be characterized by a pair of numbers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[s, \delta]</tex> such that any synchronization slip of s bits or less produces an overlap sequence differing from a legitimate codeword in at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta</tex> places. This definition is broader than that of comma freedom of index <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta</tex> , which is included as the special case of s equal to the integer part of half the code block length. For codes having the slip-detecting characteristic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[s, \delta]</tex> there exists the possibility of implementation to restore synchronism during an interval relatively free from bit errors. It is shown that certain error-correcting binary cyclic block codes can be altered to obtain the characteristic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[s, \delta]</tex> by the addition of a fixed binary vector to each codeword prior to transmission. These altered cyclic codes retain the full error-correcting power of the original cyclic codes. An error-detecting/correcting data format providing protection against the acceptance of misframed data is thus obtained without the insertion of special synchronizing sequences into the bit stream.

Introduction. Most encoding and decoding equipment operates in binary symbols, whereas it is often desirable to use a code consisting of symbols from GF(2S); each code symbol is really s binary symbols. This happens, for example, in multilevel transmission; it also happens when the chief function of the code is burst-error correction. The question discussed in this paper is when and how a binary cyclic code of block length s(2s 1) can be mapped in a way onto a cyclic code of block length 2s 1 over GF(2S). Such codes are attractive because they combine the advantages of multilevel efficiency with binary implementation; however the moral of this paper seems to be that we had better look for other ways to achieve this goal. Indeed the natural practical mapping between codes over GF(2) and codes over GF(2S) is mathematically rather unnatural. The image of a cyclic code over GF(2) may easily fail to be a linear space over GF(2S). Thus it is not surprising (although disappointing) that we appear to find only a rather small class of rather poor codes. The only previous published work on this subject known to the writer is due to M. Hanan and F. P. Palermo; these authors discuss a more general form of the same basic question, and arrive at the same basic conclusion. The plan of this paper is as follows: Section 1 describes the mapping p from the vector space over the binary field to the vector space over GF(2s), and gives an explicit form for the inverse mapping. Section 2 gives necessary and sufficient conditions for the image under (p of a binary cyclic code to be a cyclic code over GF(2S), and consequently for the image under p ` of a cyclic code over GF(2S) to be a binary cyclic code. These conditions are essentially the same as those given by Hanan and Palermo, and are definitely unhelpful from a practical point of view. Section 3 describes the class of codes, which can always be mapped back and forth, and shows that the union and intersection of an interlaced code with a mappable code is again mappable. Section 4 shows how to get new mappings from old-specifically, if a(x) is the polynomial of least degree in a mappable cyclic code (the generator polynomial), then the reciprocal polynomial of a(x) is also mappable, and so is the polynomial obtained by reversing the order of the coefficients of a(x). (The mapping function 5p is different in the three cases.) Section 5 describes the one choice of p which we know always works and shows how it works. Section 6 contains several theorems which are useful when looking for mappings; for example, we need consider only noninterlaced codes generated over GF(2s) by polynomials of degree ?s 1. The Appendix contains a series of examples, which hopefully will be useful as illustrations of the text.

The conventional parity-check matrix transformation algorithm (PTA) requires matrix inversion to transform matrices of maximum distance separable (MDS) codes. However, such matrix transformation is not always guaranteed for the class of non-MDS codes. Hence, the PTA fails for binary cyclic (BC) codes. To overcome this limitation, we developed a generalized parity-check matrix transformation (GPT) algorithm for binary cyclic codes. The GPT avoids the matrix inversion step of the PTA. It permutes the columns of the parity-check matrix based on the reliability information from the channel. Results show a significant bit error rate (BER) performance gain as compared to the existing PTA. It also presents a reasonable BER performance as compared to the other soft-decision (SD) decoding algorithms. In addition, the decoder functions within a practical decoding time complexity, particularly at the message passing stage.

The paper considers results of design and simulation of analogue-digital converters (ADC) based on current mirrors for the multi-sensor systems with parallel inputs-outputs. Such ADCs are named us as multichannel serial-parallel analog-to-digital converters based on current mirrors (M SP ADC CM). Compared with usual converters, for example reading, a bit-by-bit equilibration, and so forth, the proposed converters have a number of advantages: high speed and reliability, simplicity, small power consumption, the big degree of integration in linear and matrix structures. We discuss aspects of the design of M SP ADC CM in Gray and binary codes. It is offered, investigated and simulated the 6, 8 and more digit M SP ADC CM in Gray code and binary codes. Each channel of the overall structure consists of several base digit cells (ABC), with options for low power consumption with only one such ABC and analog memory (less than 20 CMOS transistors). Base digit cells (АВС) of such M SP ADC CM, series-pipelined in structures, consist of 20-30 CMOS transistors, one photodiode, have low (1-3.3) V supply voltage, work in current modes with the maximum values of currents (10-40) μA. Therefore such new principles of realization of high-speed low-digital M SP ADC CM have allowing, as shown by simulation experiments, to reach time of transformation less than 20-30ns at 5-8 bits of binary code and Gray code and the power consumption 1-5mW. The quantity of easily cascadable АВС depends on multi-bit ADC, and makes n, and provides quantity of quantization levels equal N=2 n . Such simple structure of M SP ADC CM with low power consumption ≤3÷5mWand supply voltage (3-7)V, and at the same time with good dynamic characteristics (frequency of digitization even for 1.5μm CMOS-technologies is 40 MHz, and can be increased up to 10 times) and accuracy (Δ quantization =156,25nA for I max =10μA ) characteristics are show. The range of optical signals, taking into account sensitivity of modern photo-detectors, can be 20-200 μW. Each channel of ADC, to reach the general power 50-100 μ W for low power consumption, can consist of only one such ABC and analog memory. To implement such serial ADC no more than 40 CMOS transistors are needed. The M SP ADC CM opens new prospects for realization linear and matrix (with picture operands) micro photo-electronic structures which are necessary for neural networks, digital optoelectronic processors, neural-fuzzy controllers, and so forth.

The important family of binary cyclic codes is explored in this chapter. Starting with cyclotomic cosets, the minimal polynomials are introduced. The Mattson–Solomon polynomial is described and it is shown to be an inverse discrete Fourier transform based on a primitive root of unity. The usefulness of the Mattson–Solomon polynomial in the design of cyclic codes is demonstrated. The relationship between idempotents and the Mattson–Solomon polynomial of a polynomial that has binary coefficients is also described. It is shown how binary cyclic codes may be easily derived from idempotents based on the cyclotomic cosets. It is demonstrated how useful this can be in the design of high-degree non-primitive binary cyclic codes. Several code examples using this construction method are presented. A table listing the complete set of the best binary cyclic codes, having the highest minimum Hamming distance, is included for all code lengths from 129 to 189 bits.

For a given binary BCH code Cn of length n = 2 s - 1 generated by a polynomial of degree r there is no binary BCH code of length (n + 1)n generated by a generalized polynomial of degree 2r. However, it does exist a binary cyclic code C (n+1)n of length (n + 1)n such that the binary BCH code Cn is embedded in C (n+1)n . Accordingly a high code rate is attained through a binary cyclic code C (n+1)n for a binary BCH code Cn . Furthermore, an algorithm proposed facilitates in a decoding of a binary BCH code Cn through the decoding of a binary cyclic code C (n+1)n , while the codes Cn and C (n+1)n have the same minimum hamming distance.

The paper considers results of design and modeling of continuously logical analog-to-digital converters (ADC) based on current mirrors for image processor and multichannel optical sensor systems with parallel inputs-outputs. For such multichannel serial-parallel analog-to-digital converters (SP ADC) it is needed base photoelectron cells, which are considered in paper. Its have a number of advantages: high speed and reliability, simplicity, small power consumption, high integration level for linear and matrix structures. We show design of the continuously logical ADC of photocurrents and its base digit cells (ABC) and its simulations. We consider CL ADC for Gray and binary codes. Each channel of the structure consists of several base digit cells (ABC) on 20-30 CMOS FETs and one photodiode. The supply voltage of the ABC is 1-3.3V, the range of an input photocurrent is 0.1 – 10μA, the transformation time is 30ns at 5-8 bit binary or Gray codes, power consumption is about 1mW. One channel of ADC with iteration is based on one ABC-3(G) and SHD, and it has only 40 CMOS transistors. The general power consumption of the ADC, in this case, is only 50-100μW, if the maximum input current is 1μA. The CL ADC opens new prospects for realization of linear and matrix image processor and photo-electronic structures with picture operands, which are necessary for neural networks, digital optoelectronic processors, neural-fuzzy controllers, and so forth.

Let s and k be integers such that s is a divisor of 2/sup k/-1. Let g(x) be a divisor of x/sup s/-1 over F/sub 2/, and let /spl pi/(x) be a primitive polynomial of degree k over F/sub 2/. We consider the binary cyclic code C of length N=2/sup k/-1 generated by (X/sup N/-1)/g(x)/spl pi/(x). For special cases, we determine the weight distribution of C by using the weights of the cyclic code of length s generated by (x/sup s/-1)/g(x). >

Constacyclic codes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="italic">uF</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>

Preface. Foreword. The ECC web site. 1. Introduction. 1.1 Error correcting coding: Basic concepts. 1.1.1 Block codes and convolutional codes. 1.1.2 Hamming distance, Hamming spheres and error correcting capability. 1.2 Linear block codes. 1.2.1 Generator and parity-check matrices. 1.2.2 The weight is the distance. 1.3 Encoding and decoding of linear block codes. 1.3.1 Encoding with G and H. 1.3.2 Standard array decoding. 1.3.3 Hamming spheres, decoding regions and the standard array. 1.4 Weight distribution and error performance. 1.4.1 Weight distribution and undetected error probability over a BSC. 1.4.2 Performance bounds over BSC, AWGN and fading channels. 1.5 General structure of a hard-decision decoder of linear codes. Problems. 2. Hamming, Golay and Reed-Muller codes. 2.1 Hamming codes. 2.1.1 Encoding and decoding procedures. 2.2 The binary Golay code. 2.2.1 Encoding. 2.2.2 Decoding. 2.2.3 Arithmetic decoding of the extended (24, 12, 8) Golay code. 2.3 Binary Reed-Muller codes. 2.3.1 Boolean polynomials and RM codes. 2.3.2 Finite geometries and majority-logic decoding. Problems. 3. Binary cyclic codes and BCH codes. 3.1 Binary cyclic codes. 3.1.1 Generator and parity-check polynomials. 3.1.2 The generator polynomial. 3.1.3 Encoding and decoding of binary cyclic codes. 3.1.4 The parity-check polynomial. 3.1.5 Shortened cyclic codes and CRC codes. 3.1.6 Fire codes. 3.2 General decoding of cyclic codes. 3.2.1 GF(2m) arithmetic. 3.3 Binary BCH codes. 3.3.1 BCH bound. 3.4 Polynomial codes. 3.5 Decoding of binary BCH codes. 3.5.1 General decoding algorithm for BCH codes. 3.5.2 The Berlekamp-Massey algorithm (BMA). 3.5.3 PGZ decoder. 3.5.4 Euclidean algorithm. 3.5.5 Chien search and error correction. 3.5.6 Errors-and-erasures decoding. 3.6 Weight distribution and performance bounds. 3.6.1 Error performance evaluation. Problems. 4. Nonbinary BCH codes: Reed-Solomon codes. 4.1 RS codes as polynomial codes. 4.2 From binary BCH to RS codes. 4.3 Decoding RS codes. 4.3.1 Remarks on decoding algorithms. 4.3.2 Errors-and-erasures decoding. 4.4 Weight distribution. Problems. 5. Binary convolutional codes. 5.1 Basic structure. 5.1.1 Recursive systematic convolutional codes. 5.1.2 Free distance. 5.2 Connections with block codes. 5.2.1 Zero-tail construction. 5.2.2 Direct-truncation construction. 5.2.3 Tail-biting construction. 5.2.4 Weight distributions. 5.3 Weight enumeration. 5.4 Performance bounds. 5.5 Decoding: Viterbi algorithm with Hamming metrics. 5.5.1 Maximum-likelihood decoding and metrics. 5.5.2 The Viterbi algorithm. 5.5.3 Implementation issues. 5.6 Punctured convolutional codes. 5.6.1 Implementation issues related to punctured convolutional codes. 5.6.2 RCPC codes. Problems. 6. Modifying and combining codes. 6.1 Modifying codes. 6.1.1 Shortening. 6.1.2 Extending. 6.1.3 Puncturing. 6.1.4 Augmenting, expurgating and lengthening. 6.2 Combining codes. 6.2.1 Time sharing of codes. 6.2.2 Direct sums of codes. 6.2.3 The |u|u + v|-construction and related techniques. 6.2.4 Products of codes. 6.2.5 Concatenated codes. 6.2.6 Generalized concatenated codes. 7. Soft-decision decoding. 7.1 Binary transmission over AWGN channels. 7.2 Viterbi algorithm with Euclidean metric. 7.3 Decoding binary linear block codes with a trellis. 7.4 The Chase algorithm. 7.5 Ordered statistics decoding. 7.6 Generalized minimum distance decoding. 7.6.1 Sufficient conditions for optimality. 7.7 List decoding. 7.8 Soft-output algorithms. 7.8.1 Soft-output Viterbi algorithm. 7.8.2 Maximum-a posteriori (MAP) algorithm. 7.8.3 Log-MAP algorithm. 7.8.4 Max-Log-MAP algorithm. 7.8.5 Soft-output OSD algorithm. Problems. 8. Iteratively decodable codes. 8.1 Iterative decoding. 8.2 Product codes. 8.2.1 Parallel concatenation: Turbo codes. 8.2.2 Serial concatenation. 8.2.3 Block product codes. 8.3 Low-density parity-check codes. 8.3.1 Tanner graphs. 8.3.2 Iterative hard-decision decoding: The bit-flip algorithm. 8.3.3 Iterative probabilistic decoding: Belief propagation. Problems. 9. Combining codes and digital modulation. 9.1 Motivation. 9.1.1 Examples of signal sets. 9.1.2 Coded modulation. 9.1.3 Distance considerations. 9.2 Trellis-coded modulation (TCM). 9.2.1 Set partitioning and trellis mapping. 9.2.2 Maximum-likelihood. 9.2.3 Distance considerations and error performance. 9.2.4 Pragmatic TCM and two-stage decoding. 9.3 Multilevel coded modulation. 9.3.1 Constructions and multistage decoding. 9.3.2 Unequal error protection with MCM. 9.4 Bit-interleaved coded modulation. 9.4.1 Gray mapping. 9.4.2 Metric generation: De-mapping. 9.4.3 Interleaving. 9.5 Turbo trellis-coded modulation. 9.5.1 Pragmatic turbo TCM. 9.5.2 Turbo TCM with symbol interleaving. 9.5.3 Turbo TCM with bit interleaving. Problems. Appendix A: Weight distributions of extended BCH codes. A.1 Length 8. A.2 Length 16. A.3 Length 32. A.4 Length 64. A.5 Length 128. Bibliography. Index.

We describe two classes of optimal binary self-complementary cyclic codes with parameters (n,2(n+2),(n-2)/2), and also two classes of optimal binary self-complementary constant-weight cyclic codes with parameters (n,2(n+1),(n-2)/2) and weight n/2 where n/spl equiv/2 (mod 4). Some of them are as good as the best codes known, and some are optimal constant-weight cyclic codes and optimal cyclic codes.

Single-track Gray codes are cyclic Gray codes with codewords of length n, such that all the n tracks which correspond to the n distinct coordinates of the codewords are cyclic shifts of the first track. We investigate the structure of such binary codes and show that there is no such code with 2/sup n/ codewords when n is a power of 2. This implies that the known codes with 2/sup n/-2n codewords. when n is a power of 2, are optimal. This result is then generalized to codes over GF(p), where p is a prime. A subclass of single-track Gray codes, called single-track Gray codes with k-spaced heads, is also defined. All known systematic constructions for single-track Gray codes result in codes from this subclass. We investigate this class and show it has a strong connection with two classes of sequences, the full-order words and the full-order self-dual words. We present an iterative construction for binary single-track Gray codes which are asymptotically optimal if an infinite family of asymptotically optimal seed-codes exists. This construction is based on an effective way to generate a large set of distinct necklaces and a merging method for cyclic Gray codes based on necklaces representatives.

This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p → 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.