Number Of Information Symbols Research Articles

Five Times Extended Reed-Solomon Codes Applicable in Memory Storage Systems

In this letter it is proved that five times extended Reed-Solomon codes contain an infinite subset of almost MDS codes with parameters: $[(q-1)+5,(q-1),5]_{GF(2^m)}$[(q-1)+5,(q-1),5]GF(2m) which are defined over a finite field $GF(2^m)$GF(2m) where $m\geq 3$m≥3 is a positive odd integer. The first three of these codes reach an upper bound for linear block code distance for the corresponding codeword lengths and number of information symbols in codewords in existing tables for optimal code parameters [1] . The relevant parts of weight spectra for the first four codes confirm the code parameters.

Nearly Optimal Constructions of PIR and Batch Codes

In this paper, we study two families of codes with availability, namely, private information retrieval (PIR) codes and batch codes . While the former requires that every information symbol has $k$ mutually disjoint recovering sets, the latter imposes this property for each multiset request of $k$ information symbols. The main problem under this paradigm is to minimize the number of redundancy symbols. We denote this value by $r_{P}(n,k)$ and $r_{B}(n,k)$ , for PIR codes and batch codes, respectively, where $n$ is the number of information symbols. Previous results showed that for any constant $k$ , $r_{P}(n,k) = \Theta (\sqrt {n})$ and $r_{B}(n,k)= {\mathcal{ O}}(\sqrt {n}\log (n))$ . In this paper, we study the asymptotic behavior of these codes for non-constant $k$ and specifically for $k=\Theta (n^\epsilon)$ . We also study the largest value of $k$ such that the rate of the codes approaches 1 and show that for all $\epsilon , $r_{P}(n,n^\epsilon) = o(n)$ and $r_{B}(n,n^\epsilon) = o(n)$ . Furthermore, several more results are proved for the case of fixed $k$ .

Linear codes and some their applications

The application of various areas of the theory of coding in the problems of analysis of the genome includes different areas. For instance, the method for detecting linear block codes, which takes into account possible insertions and divisions in DNA sequences. For a lot of such problems, we shall consider the number of codes of the required dimension. One of the most important such tasks is counting their number. We study this area it in this paper. We solve the problem of defining the number of cyclic codes over finite field with two arbitrary fixed parameters. These parameters are n, i.e., the length of the code, and k, i.e., the number of information symbols. The algorithm to solve this problem in general is described in this paper. Also the results at some fixed values are given. We have counted some constants connected with coding theory, which can be used in some problems of genome analysis.

Systematic Error-Correcting Codes for Permutations and Multi-Permutations

Multi-permutations and in particular permutations appear in various applications in an information theory. New applications, such as rank modulation for flash memories, have suggested the need to consider error-correcting codes for multi-permutations. In this paper, we study systematic error-correcting codes for multi-permutations in general and for permutations in particular. For a given number of information symbols $k$ , and for any integer $t$ , we present a construction of ${(k+r,k)}$ systematic $t$ -error-correcting codes, for permutations of length $k+r$ , where the number of redundancy symbols $r$ is relatively small. In particular, for a given $t$ and for sufficiently large $k$ , we obtain $r=t+1$ , while a lower bound on the number of redundancy symbols is shown to be $t$ . The same construction is also applied to obtain related systematic error-correcting codes for any types of multi-permutations.

Distributed Space-Time Codes for Full-Duplex IR-UWB Amplify-and-Forward Cooperation

In this paper, we consider the problem of full-duplex (FD) relaying in the context of impulse-radio ultra-wideband (IR-UWB) communications. In particular, we propose two novel distributed space-time block codes (STBCs) suitable for the amplify-and-forward (AF) cooperation protocol with one and two relays. Despite the fact that FD relaying results in significant levels of interference between the transmit and receive antennas of each relay, it introduces new concepts to the problem of distributed STBC design. Compared to half-duplex (HD) STBC, FD-STBC is subject to an additional constraint that imposes the structure of the codewords while it offers the predominant advantage that resides in the possibility of including a smaller number of information symbols per codeword for achieving a full rate. We take advantage of this potential for constructing fully-diverse, full-rate and totally-real IR-UWB FD-STBCs that outperform the existing HD-STBCs for all practical levels of the residual self loop interference. In fact, for the sake of transmitting at full rate, the best known distributed HD-STBCs for the non-orthogonal AF protocol require the joint encoding/decoding of $4N_{r}$ symbols where $N_{r} $ is the number of relays. On the other hand, the proposed FD-STBCs require embedding only $N_{r} +2$ symbols per codeword for transmitting at full rate. Following from this fact, not only the decoding complexity is reduced, but also the coding gains are improved resulting in enhanced performance levels.

A Matroidal Framework for Network-Error Correcting Codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. A particularly interesting feature of this development is the ability to construct (scalar and vector) linearly solvable networks using certain classes of matroids. Furthermore, it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios. The current work attempts to establish a connection between matroid theory and network-error correcting and detecting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of linear network-error detecting codes to arrive at the definition of a matroidal error detecting network (and similarly, a matroidal error correcting network abstracting from network-error correcting codes). An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error detecting (correcting) network code if and only if it is a matroidal error detecting (correcting) network associated with a representable matroid. Therefore, constructing such network-error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa. We then present algorithms that enable the construction of matroidal error detecting and correcting networks with a specified capability of network-error correction. Using these construction algorithms, a large class of hitherto unknown scalar linearly solvable networks with multisource, multicast, and multiple-unicast network-error correcting codes is made available for theoretical use and practical implementation, with parameters, such as number of information symbols, number of sinks, number of coding nodes, error correcting capability, and so on, being arbitrary but for computing power (for the execution of the algorithms). The complexity of the construction of these networks is shown to be comparable with the complexity of existing algorithms that design multicast scalar linear network-error correcting codes. Finally, we also show that linear network coding is not sufficient for the general network-error correction (detection) problem with arbitrary demands. In particular, for the same number of network errors, we show a network for which there is a nonlinear network-error detecting code satisfying the demands at the sinks, whereas there are no linear network-error detecting codes that do the same.

Random Matrices and Codes for the Erasure Channel

The design of erasure correcting codes and their decoding algorithms is now at the point where capacity achieving codes are available with decoding algorithms that have complexity that is linear in the number of information symbols. One aspect of these codes is that the overhead (number of coded symbols beyond the number of information symbols required to achieve decoding completion with high probability) is linear in k. This work considers a new class of random codes which have the following advantages: (i) the overhead is constant (in the range of 5 to 10), independent of the number of data symbols being encoded (ii) the probability of completing decoding for such an overhead is essentially one (iii) the codes are effective for a number of information symbols as low as a few tens (iv) the only probability distribution required is the uniform distribution. The price for these properties is that the decoding complexity is greater, on the order of k 3/2. However, for the lower values of k where these codes are of particular interest, this increase in complexity might be outweighed by their advantages. The parity check matrices of these codes are chosen at random as windowed matrices, i.e. for each column an initial starting position of a window of length w is chosen and the succeeding w positions are chosen at random as zero or one. It can be shown that it is necessary that w=O(k 1/2) for the probabilistic matrix rank properties to behave as a non-windowed random matrix. The sufficiency of the condition has so far been established by extensive simulation, although other arguments strongly support this conclusion. The properties of the codes described depend heavily on the rank properties of random matrices over finite fields. Known results on such matrices are briefly reviewed and several conjectures needed in the discussion of the code properties, are stated. The likelihood of the validity of the conjectures is supported through extensive experimentation. Mathematical proof of the conjectures would be of great value for their own interest as well of the particular coding application described here.

Efficient encoding for a class of subspace subcodes

Let S consist of all words of a code C for which each symbol is in a stipulated subalphabet, possibly different for distinct positions. We consider the special case where C is a linear maximum-distance separable (MDS) code, and the subalphabets are linear subspaces over the ground field with equal dimensions. We give an explicit algorithm for selecting the subspaces in such a way that a straightforward systematic encoding algorithm, based on an encoder for C, can be applied. The number of information symbols that can be encoded with this algorithm equals a well-known lower bound on the dimension of S.

Exponential sums and Goppa codes. II

For pt.I, see Proc. AMS, vol.III, p.523-31 (1991). The minimum distance of a Goppa code is found when the length of code satisfies a certain inequality on the degree of the Goppa polynomial. In order to do this, conditions are improved on a theorem of E. Bombieri (1966). This improvement is used also to generalize a previous result on the minimum distance of the dual of a Goppa code. This approach is generalized and results are obtained about the parameters of a class of subfield subcodes of geometric Goppa codes; in other words, the covering radii are estimated, and further, the number of information symbols whenever the minimum distance is small in relation to the length of the code is found. Finally, a bound on the minimum distance of the dual code is discussed. >

New multilevel codes over GF(q)

Set partitioning is applied to multidimensional signal spaces over GF(q), i.e., GF/sup n1/(q) (n1<or=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/sub min/>or=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.<<ETX>>

A hybrid ARQ scheme with adaptive forward error correction for satellite communications

A hybrid automatic-repeat-request system combined with adaptive forward error correction (ARQ/AFEC) which adaptively alters the error-correcting capability according to channel state using a nonsystematic Reed-Solomon code is presented. lt is shown that the system can alter the error-correcting capability by varying the number of information symbols fed to the decoder only, and not change the hardware of the encoder. Performance evaluations indicate that the ARQ/AFEC system has much higher frame efficiency than ARQ without AFEC for low-bit error rate and has high frame efficiency in a wide range of bit error rates. >

A versatile time-domain Reed-Solomon decoder

A versatile Reed-Solomon (RS) decoder structure based on the time-domain decoding algorithm (transform decoding without transforms) is developed. The algorithm is restructured, and a method is given to decode any RS code generated by any generator polynomial. The main advantage of the decoder structure is its versatility, that is, it can be programmed to decode any Reed-Solomon code defined in Galois field (GF) 2/sup m/ with a fixed symbol size m. This decoder can correct errors and erasures for any RS code, including shortened and singly extended codes. It is shown that the decoder has a very simple structure and can be used to design high-speed single-chip VLSI decoders. As an example, a gate-array-based programmable RS decoder is implemented on a single chip. This decoder chip can decode any RS code defined in GF (2/sup 5/) with any code word length and any number of information symbols. The decoder chip is fabricated using low-power 1.5- mu , two-layer-metal, HCMOS technology. >

Packet error probabilities in frequency-hopped spread-spectrum packet radio networks-memoryless frequency-hopping patterns considered

The packet error probability induced in a frequency-hopped spread-spectrum packet radio network is computed. The frequency spectrum is divided into q frequency bins. Each packet is exactly one codeword from an (M, L) Reed-Solomon code (M=number of codeword symbols (bytes); L=number of information symbols (bytes)). Every user in the network sends each of the M bytes of his packet at a frequency chosen among the q frequencies with equal probability and independently of the frequencies chosen for other bytes (i.e., memoryless frequency-hopping patterns). Statistically independent frequency-hopping patterns correspond to different users in the network. Provided that K users have simultaneously transmitted their packets on the channel and a receiver has locked on to one of these K packets, the probability that this packet is not decoded correctly is evaluated. It is also shown that although memoryless frequency-hopping patterns are utilized, the byte errors at the receiver are not statistically independent; instead they exhibit a Markovian structure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Synchronous Bounded Delay Coding for Input Restricted Channels

Consider the class of input restricted channels whose constraints can be modelled by finite state machines. This paper treats the problem of constructing fixed rate (synchronous) codes for such channels subject to a constraint that the coding delay be bounded by a parameter M. M is the maximum number of information symbols required by the coder in choosing a word during transmission.

On channels and codes for the Lee metric

Almost all nonbinary codes were designed for the Hamming metric. The Lee metric was defined by Lee in 1958. Golomb and Welch (1968) and Berlekamp, 1968a, Berlekamp, 1968b have designed codes for the Lee metric. In this paper, we derive all the discrete, memoryless, symmetric channels matched to the Lee metric, and investigate general properties of Lee metric block codes. Finally, a class of cyclic Lee metric codes is defined and the number of information symbols is discussed.

New generalizations of the Reed-Muller codes--II: Nonprimitive codes

In this paper a class of nonprimitive cyclic codes quite similar in structure to the original Reed-Muller codes is presented. These codes, referred to herein as nonprimitive Reed-Muller codes, are shown to possess many of the properties of the primitive codes. Specifically, two major results are presented. First the code length, number of information symbols, and minimum distance are shown to be related by means of a parameter known as the order of the code. These relationships show that for given values of code length and rate the codes have relatively large minimum distances. It is also shown that the codes are subcodes of the BCH codes of the same length and guaranteed minimum distance; thus in general the codes are not as powerful as the BCH codes. However, for most interesting values of code length and rate the difference between the two types of codes is slight. The second result is the observation that the codes can be decoded with a variation of the original algorithm proposed by Reed for the Reed-Muller codes. In other words, they are L -step orthogonalizable. Because of their large minimum distances and the simplicity of their decoders, nonprimitive Reed-Muller codes seem attractive for use in error-control systems requiring multiple random-error correction.

The Enumeration of Information Symbols in BCH Codes

This paper presents certain formulas for I(q, n, d), the number of information symbols in the q-ary Bose-Chaudhuri-Hocquenghem code of block length n = qm − 1 and designed distance d. By appropriate manipulations on the m-digit q-ary representation of d, we derive a simple linear recurrence for a sequence whose mth term is the number of information symbols in the BCH code. In addition to an exact solution of all finite cases, we obtain exact asymptotic results, as n and d go to infinity while their ratio n/d remains fixed. In this limit, the number of information symbols increases as n'. Specifically, we show that for fixed u, 0 ≦ u ≦ 1, $\lim\limits_{m \rightarrow \infty} q^{-ms} I(q,q^{m} - 1, uq^{m})=1$ where s is a singular function of u. The function s(u) is continuous and monotonic nonincreasing; it has derivative zero almost everywhere. Yet s(0) = 1 and s(1) = 0. For q = 2, s(u) is plotted in Fig. 1.