A Theoretical Method for Estimating Performance of Reed-Solomon Codes Concatenated with Orthogonal Space-Time Block Codes

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Reduction of PAPR in coded OFDM using fast Reed-Solomon codes over prime Galois fields

In this work, two new techniques using Reed–Solomon (RS) codes over GF(257) and GF(65,537) are proposed for peak-to-average power ratio (PAPR) reduction in coded orthogonal frequency division multiplexing (OFDM) systems. The lengths of these codes are well-matched to the length of OFDM frames. Over these fields, the block lengths of codes are powers of two and we fully exploit the radix-2 fast Fourier transform algorithms. Multiplications and additions are simple modulus operations. These codes provide desirable randomness with a small perturbation in information symbols that is essential for generation of different statistically independent candidates. Our simulations show that the PAPR reduction ability of RS codes is the same as that of conventional selected mapping (SLM), but contrary to SLM, we can get error correction capability. Also for the second proposed technique, the transmission of side information is not needed. To the best of our knowledge, this is the first work using RS codes for PAPR reduction in single-input single-output systems.     

Fast technique for computing syndromes of B.C.H. and Reed-Solomon codes

A new scheme for reducing the numerical complexity of the standard B.C.H. and Reed?Solomon (R.S.) decoding algorithms is developed. Specifically, the process of calculating syndromes over GF(2m) is shown to require only a small fraction of the number of multiplications and additions that is required by using standard methods. As an example, the calculation of the 32 syndromes of the (255, 223, 33) Reed?Solomon code (NASA standard for concatenation with convolutional codes) is shown to require 90% fewer multiplications and 78% fewer additions than the conventional method of computation. A computer simulation also verifies these results.     

Low-density parity check codes over GF(q)

Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10^-5 at E_b/N₀=0.2 dB     

Serial concatenation of Hermitian codes with ring‐TCM codes

Hermitian codes are a class of very long algebraic‐geometric (AG) codes constructed from Hermitian curves, which outperform Reed–Solomon codes defined over the same finite fields and with the same code rates, as recently demonstrated by the authors. However, since there are no soft‐decision decoding algorithms for AG codes in the literature, the performance of Hermitian codes is limited and their potential is yet to be realized. An alternative method for achieving more significant coding gains is presented in this paper by serially concatenating long Hermitian codes with ring‐trellis‐coded modulation codes over the ring of integers ?₄ and evaluating their performance through simulation results on the AWGN and Rayleigh fading channels. The scheme achieves large coding gains over single Hermitian and Reed–Solomon codes with no increase in bandwidth use and a performance comparable with the well‐known capacity‐approaching codes. Copyright © 2007 John Wiley & Sons, Ltd.     

A Framework for the Construction ofGolay Sequences

In 1999, Davis and Jedwab gave an explicit algebraic normal form for m!/2 - 2^h(m+1) ordered Golay pairs of length 2^mmiddot over Z₂h, involving m!/2 - 2^h(m+1) Golay sequences. In 2005, Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of a "crossover" of the aperiodic autocorrelation function of certain quaternary length eight sequences belonging to Golay pairs, and that they spawn further new quaternary Golay sequences and pairs of length 2^m for m > 4 under Budisin's 1990 iterative construction. The total number of Golay sequences and pairs spawned in this way is counted, and their algebraic normal form is given explicitly. A framework of constructions is derived in which Turyn's 1974 product construction, together with several variations, plays a key role. All previously known Golay sequences and pairs of length 2^m over Z₂h can be obtained directly in explicit algebraic normal form from this framework. Furthermore, additional quaternary Golay sequences and pairs of length 2^m are produced that cannot be obtained from any other known construction. The framework generalizes readily to lengths that are not a power of 2, and to alphabets other than Z_2h .     

Algorithm for recursively generating irreducible polynomials

A simple method for checking the irreducibility of f(x²+x+1) and f(x²+x) over GF(2) is described. An algorithm that recursively generates irreducible polynomials is presented. These irreducible polynomials are useful in constructing finite fields for applications in error-correcting codes and cryptography     

On the probability of symbol error in Viterbi decoders

An upper bound is derived on the probability that at least one of a sequence of B consecutive bits at the output of a Viterbi (1979) decoder is in error. Such a bound is useful for the analysis of concatenated coding schemes employing an outer block code over GF(2^B) (typically a Reed-Solomon (RS) code), an inner convolutional code, and a symbol (GF(2^B)) interleaver separating the two codes. The bound demonstrates that in such coding schemes a symbol interleaver is preferable to a bit interleaver. It also suggests a new criterion for good inner convolutional codes     

High-Rate Nonbinary Regular Quasi-Cyclic LDPC Codes for Optical Communications

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10^-12. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.     

Exact lower bounds on the codelength of three-steppermutation-decodable cyclic codes

The exact lower bounds on codelength n for three-step ( T, U) permutation decodable binary cyclic codes of even-valued error number t (t⩾4) are presented. Since the derivation of these results involves only the error position, the results are applicable to cyclic codes over GF(2^m)     

A rate 4/6 (d=1, k=11) block-decodable runlength-limited code

A new rate 4/6 (d=1, k'=11) runlength-limited code which is well adapted to byte-oriented storage systems is presented. The new code has the virtue that it can be decoded on a block basis, i.e., without knowledge of previous or next codewords, and, therefore, it does not suffer from error propagation. This code is particularly attractive as many commercially available Reed-Solomon codes operate in GF(2⁸ )     

New array codes for multiple phased burst correction

An optimal family of array codes over GF(q) for correcting multiple phased burst errors and erasures, where each phased burst corresponds to an erroneous or erased column in a code array, is introduced. As for erasures, these array codes have an efficient decoding algorithm which avoids multiplications (or divisions) over extension fields, replacing these operations with cyclic shifts of vectors over GF(q). The erasure decoding algorithm can be adapted easily to handle single column errors as well. The codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array, thus generalizing a previously known construction for the special case of two erasures. Algebraically, they can be interpreted as Reed-Solomon codes. When q is primitive in GF(q), the resulting codes become (conventional) Reed-Solomon codes of length P over GF(q^p-1), in which case the new erasure decoding technique can be incorporated into the Berlekamp-Massey algorithm, yielding a faster way to compute the values of any prescribed number of errors     

Architecture de turbo-décodeur en blocs entièrement parallèle pour la transmission de données au-delà du Gbit/s

This paper presents a new circuit architecture for turbo decoding, which achieves ultra high data rates when using product codes as error correcting codes. This architecture is able to decode product codes using binary BCH or m-ary Reed Solomon component codes. The major advantage of our full-parallel architecture is that it enables the memory block between each half-iteration to be removed. In fact, the proposed architecture opens the way to numerous applications such as optical transmission. In particular, our block turbo decoding architecture can support optical transmission at data rates above Gbit/s     

Orthogonality of binary codes derived from Reed-Solomon codes

The author provides a simple method for determining the orthogonality of binary codes derived from Reed-Solomon codes and other cyclic codes of length 2^m-1 over GF(2^m) for m bits. Depending on the spectra of the codes, it is sufficient to test a small number of single-frequency pairs for orthogonality, and a pair of bases may be tested in each case simply by summing the appropriate powers of elements of the dual bases. This simple test can be used to find self-orthogonal codes. For even values of m, the author presents a technique that can be used to choose a basis that produces a self-orthogonal, doubly-even code in certain cases, particularly when m is highly composite. If m is a power of 2, this technique can be used to find self-dual bases for GF(2 ^m). Although the primary emphasis is on testing for self orthogonality, the fundamental theorems presented apply also to the orthogonality of two different codes     

New multilevel codes over GF(q)

Set partitioning is applied to multidimensional signal spaces over GF(q), i.e., GFⁿ¹(q) (n1⩽q ), and it is shown how to construct both multilevel block codes and multilevel trellis codes over GF(q). Multilevel (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum distance d_min⩾d are presented. These codes use Reed-Solomon codes as component codes. Longer multilevel block codes are also constructed using q-ary block codes with block length longer than q+1 as component codes. Some quaternary multilevel block codes are presented with the same length and number of information symbols as, but larger distance than, the best previously known quaternary one-level block codes. It is proved that if all the component block codes are linear. the multilevel block code is also linear. Low-rate q-ary convolutional codes, word-error-correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multilevel trellis codes over GF(q) or binary-to-q-ary trellis codes     

Cyclic block codes definable over multiple finite fields

It is shown that there exists a class of error correcting codes which can be viewed both as binary cyclic codes and as multilevel cyclic codes. This class contains not only the Burton codes but also some Reed Solomon codes. The conditions which codes of this class must satisfy are explained and some simple examples are given     

A better channel coding for a Link‐16 waveform in pulsed‐noise interference

Link‐16 is a tactical data link currently used by North Atlantic Treaty Organization (NATO) countries, the United States and its allies. The Link‐16 waveform features Reed–Solomon codes for channel coding, cyclic code‐shift keying for 32‐ary baseband symbol modulation, minimum‐shift keying for waveform modulation, and frequency hopping for transmission security. In addition to the original errors‐only decoding of Reed–Solomon codes, both an errors‐and‐erasures decoding (EED) and a special concatenated coding are proposed in this paper to determine a better channel coding scheme for a Link‐16 waveform with noncoherent detection in the presence of pulsed‐noise interference (PNI). The investigation is first carried out both analytically and by simulation for the original Link‐16 waveform transmitted over AWGN. It is then accomplished analytically for the proposed waveforms in both AWGN and PNI. The results show that EED achieves the best error rate performance for a Link‐16 waveform in both AWGN and PNI when the signal‐to‐noise ratio is relatively small. When both the signal‐to‐noise ratio is sufficiently large and the fraction active time of PNI is small, the proposed concatenated coding outperforms both EED and errors‐only decoding. Copyright © 2012 John Wiley & Sons, Ltd.     

Bounds on the bit error probability of a linear cyclic code overGF(2l) and its extended code

An upper bound on the bit-error probability (BEP) of a linear cyclic code over GF(2^l) with hard-decision (HD) maximum-likelihood (ML) decoding on memoryless symmetric channels is derived. Performance results are presented for Reed-Solomon codes on GF(32), GF(64), and GF(128). Also, a union upper bound on the BEP of a linear cyclic code with either hard- or soft-decision ML decoding is developed, as well as the corresponding bounds for the extended code of a linear cyclic code. Using these bounds, which are tight at low bit error rate, the performance advantage of soft-decision (SD) ML and HD ML over bounded-distance (BD) decoding is established     

A New Family of Ternary Almost Perfect Nonlinear Mappings

A mapping f(x) from GF(pⁿ) to GF(pⁿ) is differentially k-uniform if k is the maximum number of solutions x isin GF(pⁿ) of f(x+a) - f(x) = b, where a, b isin GF(pⁿ) and a ne 0. A 2-uniform mapping is called almost perfect nonlinear (APN). This correspondence describes new families of ternary APN mappings over GF(3ⁿ), n>3 odd, of the form f(x) = ux^d + x^{d 2} where d₁ = (3^n-1)/2 - 1 and d₂ = 3ⁿ - 2.