S.B.Wicker and V.K.Bhargava, Reed Solomon Codes and their Applications. IEEE Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an error. Hence, it is desirable that the data hiding scheme be adjusted in such a manner that there are very few (or no) insertions. Thus, using a good erasure correcting code, one can deal with insertions deletions without significant decline in original embedding rate. Reed Solomon codes [8] have been used for ET scheme and Repeat Accumulate codes [9] have been used for the SEC scheme as described in following sections. B. Reed Solomon (RS) coding for ET scheme Reed Solomon codes [8] are MDS codes, such that any k coordinates of an (n,k) RS code can be treated as message positions. ....

....with insertions deletions without significant decline in original embedding rate. Reed Solomon codes [8] have been used for ET scheme and Repeat Accumulate codes [9] have been used for the SEC scheme as described in following sections. B. Reed Solomon (RS) coding for ET scheme Reed Solomon codes [8] are MDS codes, such that any k coordinates of an (n,k) RS code can be treated as message positions. In other words, the k message symbols can be decoded even if any k symbols out of total n symbols sent are received at the decoder. The code can correct (n k) erasures, or half as many errors. The ....

S. B. Wicker and V. K. Bhargava, Reed-Solomon Codes and Their Applications, IEEE Press, 1994.

....through di#erent colored filters. In the Voyager series after the late 70s the binary extended Golay code 24 , a [24, 12, 8] code, was used in the color photography. Later on in the Voyager series of spacecraft a di#erent type of codes, viz. convolutional Reed Solomon codes, was used: see [17]. We describe the Reed Solomon codes in Section 14.5.2. 14.4.2 First order Reed Muller codes A full account of the Reed Muller codes, which are all binary codes, can be found in [10] or [1, Chapter 5] We describe here simply a way to construct the [32, 6, 16] code used for space photography, ....

....for 24 is given in [16, Chapter 4] The perfect binary Golay code 23 may be obtained from 24 by deleting any coordinate. 14.5 Compact discs and Reed Solomon codes A full account of the use of Reed Solomon codes for error correction in compact discs is given in [7] 16, Chapter 7] or [17]. Sound is stored on a compact disc by dividing it up into small parts and representing these parts by binary data, just as pictures are divided up, as described in Section 14.4. A compact disc is made by sampling sound waves 44,100 times per second, the amplitude measured and assigned a value ....

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Stephen B. Wicker and Vijay K. Bhargava (Editors). Reed-Solomon Codes and their Applications. IEEE Press, 1994.

....codes that can be found in a wide variety of digital communications systems, from digital media to wireless communications and deepspace probes. The ubiquitous nature of these codes continues to fuel research into decoding algorithms some forty years after their introduction (see e.g. [1]) Reed Solomon codes have been employed in a wide spectrum of digital communications systems because they provide powerful error correction with only a small number of overhead symbols. Reed Solomon codewords consist of non binary symbols and therefore the correction of a single symbol could ....

S. B. Wicker and V. K. Bhargava, eds., Reed-Solomon Codes and Their Applications. New York, New York: IEEE Press, 1994. 18

....an error. Hence, it is desirable that the data hiding scheme be adjusted in such a manner that there are very few (or no) insertions. Thus, using a good erasure correcting code, one can deal with insertions deletions without significant decline in original embedding rate. Reed Solomon codes [8] have been used for ET scheme and Repeat Accumulate codes [9] have been used for the SEC scheme as described in following sections. B. Reed Solomon (RS) coding for ET scheme Reed Solomon codes [8] are MDS codes, such that any k coordinates of an (n,k) RS code can be treated as message positions. ....

....with insertions deletions without significant decline in original embedding rate. Reed Solomon codes [8] have been used for ET scheme and Repeat Accumulate codes [9] have been used for the SEC scheme as described in following sections. B. Reed Solomon (RS) coding for ET scheme Reed Solomon codes [8] are MDS codes, such that any k coordinates of an (n,k) RS code can be treated as message positions. In other words, the k message symbols can be decoded even if any k symbols out of total n symbols sent are received at the decoder. The code can correct (n k) erasures, or half as many errors. The ....

S.B. Wicker and V. K. Bhargava, Reed-Solomon Codes and Their Applications, IEEE Press, 1994.

....through di erent colored lters. In the Voyager series after the late 70s the binary extended Golay code G 24 , a [24; 12; 8] code, was used in the color photography. Later on in the Voyager series of spacecraft a di erent type of codes, viz. convolutional Reed Solomon codes, was used: see [17]. We describe the Reed Solomon codes in Section 14.5.2. 14.4.2 First order Reed Muller codes A full account of the Reed Muller codes, which are all binary codes, can be found in [10] or [1, Chapter 5] We describe here simply a way to construct the [32; 6; 16] code used for space photography, ....

....in [16, Chapter 4] The perfect binary Golay code G 23 may be obtained from G 24 by deleting any coordinate. 14.5 Compact discs and Reed Solomon codes 14.5. 1 Introduction A full account of the use of Reed Solomon codes for error correction in compact discs is given in [7] 16, Chapter 7] or [17]. Sound is stored on a compact disc by dividing it up into small parts and representing these parts by binary data, just as pictures are divided up, as described in Section 14.4. A compact disc is made by sampling sound waves 44,100 times per second, the amplitude measured and assigned a value ....

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Stephen B. Wicker and Vijay K. Bhargava (Editors). Reed-Solomon Codes and their Applications. IEEE Press,

....recorder (see Fig. 1) the channel encoder translates source data plus error correction (parity check) symbols into a sequence that complies with the channel constraints imposed. In virtually all recording systems in the market, the error correction code (ECC) is based on Reed Solomon codes [1]. There are, however, many flavors in channel codes. Notably, spectral shaping and runlength limited codes have found widespread usage in consumer type recording systems such as Compact Disc, DAT, DVC, etc. 2] Since only a limited repertoire of sequences is used, the rate of a code that ....

S. B. Wicker and V. K. Bhargava, Eds., Reed-Solomon Codes and Their Applications. New York: IEEE Press, 1994.

....tracking. As the reading of disks is virtually noiseless, other properties, such as, for example, robustness against additive noise play a minor role. Read errors are mostly caused by imperfections of the disk and can be resolved by sophisticated error burst correcting Reed Solomon codes [7]. We start, in the next sections, with an outline of the properties of DCRLL sequences. Thereafter, it will be shown how practical codes can be devised that satisfy the given channel constraints. It will be shown, among others, that industry standard RLL codes can be supplemented by a simple ....

S. B. Wicker and V. K. Bhargava, Eds., Reed--Solomon Codes and Their Applications. Piscataway, NJ: IEEE Press, 1994.

....in the worst case visiting all the codewords of which there are q k many. The aim is to design more efficient algorithms for special classes of codes. As an example, consider the Reed Solomon codes, one of the most important classes of codes both from a theoretical and a practical point of view [16]. Fix n different elements x 1 ; x n of the field F q , and suppose that k n are positive integers. The image of the homomorphism fl mapping a polynomial f of degree less than k to the vector (f(x 1 ) f(x n ) is a linear subspace of F n q , hence a linear code. Using the fact ....

S. Wicker and V. Bhargava, eds., Reed-Solomon Codes and their Applications, IEEE Press, 1994.

....decreases the number of symbol errors in each RS Code block. When the number of symbol errors in a RS Code block is below 8, a (255,239) RS Code can decode the block without error. 5.3.3 Soft decision All simulations for these concatenated codes have used hard decisions between the two codes. In [11,19] a soft output Viterbi decoder is presented which provides reliability information to an outer RS decoder. The same structure could be applied to the concatenated scheme presented in this chapter. The soft decisions would further improve the performance. 5.3.4 Variable rate RS codes [6, 10] ....

S. B. Wicker and V. K. Bhargava, "Reed-Solomon codes and their applications," IEEE Press, 1994. 63

....held in Trondheim, Norway, and in the Ph.D. thesis of the first author. 0 1 Introduction Galois fields have gained wide spread applications in modern communication systems such as computer networks, satellite links, or compact disks. They all use finite field arithmetics for error correction [32] or for cryptographic algorithms [22] Both issues are central for the provision of reliable and secure communication. Further applications can be found in signal processing [3] and pseudo random number generation [30] Modern applications in many cases call for VLSI implementations of the ....

S.B. Wicker and V.K. Bhargava, editors. Reed-Solomon Codes and Their Applications. IEEE Press, 1994. 21

....use the Low Gain Antenna (LGA) which will reduce the available data rate from 134,400 bits s to at most 160 bits s. The currently planned coding scheme for the LGA is a memory 13, rate 1 4 convolutional code concatenated with a time varying Reed Solomon code achieving an overall rate of 223.5 255 [1]. The planned coding scheme for the LGA will achieve a Bit Error Ratio (BER) of 10 #7 at an E b #N 0 of 0.6 dB. Part of the memory 13 convolutional code is formed from a NASA standard memory 6, rate 1 2 convolutional code. This memory 6 non systematic encoder is directly hardwired to the LGA ....

....product of the input bits and the coder impulse response. The coder response is defined by its binary generating polynomials G 1 and G 2 . Let b k be the input at time k and d i k , i # 0, 1 the output. Thus, we have d i k # # # 1 j#0 G i j b k j (3) 8 where G 1 # [1, 0, 1] and G 2 # [1, 1, 1] are the binary sequences which generate the code depicted in the Figure 3 and the sum represents the modulo 2 summation (x y # x # y) We can also express these relations in an algebraic way: d i k # G i # S k (4) where # is the binary dot product operator and S ....

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S. B. Wicker and V. K. Bhargava, "Reed Solomon codes and their application," chapter 3, section 4, 1994 IEEE Press.

.... 1 , q d , d] For example, NASA uses a [255 , 223 , 33] RS code over GF(2 8 ) for deep space communication. In another widely used application, a [32 , 28 , 5] code and a [28 , 24 , 5] code are used in an interleaved scheme to correct burst errors of length up to 4000 bits on compact discs [2]. In a sense, Dick s 1950 paper set off the current avalanche of coding theory and applications, although he left it to others to ride the avalanche. Numerical analysis Much of Dick s early work was in numerical analysis. One of his contributions to this field was the Hamming predictor corrector ....

S. B. WICKER, V. K. BHARGAVA, eds., ReedSolomon Codes and Their Applications, IEEE Press, New York, 1994.

....coding system was obtained with repeated Viterbi decoding trials with forced states [69] Trellis pruning and the use of an outer RS code with non uniform correcting capability was described in [71, 72] The drawback of this method is that the optimal code profile depends on the E b N 0 used. In [73, 74] a soft output Viterbi decoder is presented which can provide reliability information to the outer RS decoder while accepting reliability information from a previous RS decoding step in an interleaved scheme. This can be explained using Figure 7.2 as in [74] We assume that after the first ....

....depends on the E b N 0 used. In [73, 74] a soft output Viterbi decoder is presented which can provide reliability information to the outer RS decoder while accepting reliability information from a previous RS decoding step in an interleaved scheme. This can be explained using Figure 7. 2 as in [74]. We assume that after the first decoding pass two RS codewords are successfully decoded. For the second iteration, the soft output Viterbi decoder is forced through known correct states. This reduces the BER for the inner code which in turn will lead to more RS codewords correctly decoded and so ....

S. B. Wicker and V. K. Bhargava, Reed--Solomon codes and their applications, New York, IEEE Press, 1994.

....that it yields a (quadratic time) MDS code that runs in real time for medium quality video transmission on existing workstations [2, 1] i.e. at the rate of a few megabits per second. The version of Reed Solomon codes we use is based on Cauchy matrices over finite fields (see for example [6] and [11]) Every square submatrix of a Cauchy matrix is invertible. Therefore Cauchy matrices can be used to implement linear, systematic codes, i.e. the encoding is a linear function of the message and the unencoded message is part of the encoding. Since typically the encoding size is only a constant ....

....As required by Theorem 2.2, every square submatrix of a Cauchy matrix is nonsingular. Cauchy matrices are easier to invert than general matrices. This is important to achieve efficient decoding algorithms. MDS codes based on Cauchy matrices are a variant of Reed Solomon codes (see for example [11]) We will describe Cauchy matrices in detail in Section 5. In practice it turns out to be more efficient to define packets as containing b words consisting of w bits each than to define them in terms of single bits. In our implementations we chose w to be 32; the word size on a Sun SPARCstation. ....

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S. B. Wicker, V. K. Bhargava, Reed-Solomon Codes and their Applications , IEEE Press, New York, 1994.

....and Reed Solomon codes, hence the labeling of the technique as Reed Solomon coding. The technique has recently been discussed in varying levels of detail by Gibson [Gib92] Schwarz [SB92] and Burkhard [BM93] with citations of standard texts on error correcting codes [Ber68, PW72, MS77, vL82, WB94] for completeness. There is one problem with all the above discussions of this technique they require the reader to have a decent knowledge of algebra and coding theory. Any programmer with a bachelor s degree in computer science has the skills to implement this technique, however few such ....

S. B. Wicker and V. K. Bhargava. Reed-Solomon Codes and Their Applications. IEEE Press, New York, 1994.

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S.B.Wicker and V.K.Bhargava, Reed Solomon Codes and their Applications. IEEE Press, 1994.

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S. B. Wicker and V. K. Bhargava, Reed--Solomon Codes and Their Applications. New York: IEEE Press, 1994.

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S. B. Wicker and V. K. Bhargava, Reed--Solomon Codes and Their Applications. New York: IEEE Press, 1994.

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S. B. Wicker and V. K. Bhargava, "Reed-Solomon Codes and Their Applications,"Wiley-IEEE Press,, September 1999.

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S. B. Wicker and V. K. Bhargava (Eds), Reed-Solomon Codes and Their Applications, IEEE Press, 1994.

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S.Wicker, V.K.Bhargava Ed., Reed-Solomon Codes and Their Applications, IEEE Press, 1994

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S. Wicker and V. Bhargava, Reed-Solomon Codes and Their Applications. IEEE Press, 1994.

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S. B. Wicker and V. K. Bhargava (Eds), Reed-Solomon Codes and Their Applications, IEEE Press, 1994.

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M. A. Hasan, V. K. Bhargava, and T. Le-Ngoc, Reed--Solomon Codes and Their Applications, S. B. Wicker and V. K. Bhargava, Eds. Piscataway, NJ: IEEE Press, 1994. Algorithms and architectures for a VLSI Reed--Solomon codec.

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S. B. Wicker and V. K. Bhargava, Reed-Solomon Codes and Their Applications, IEEE Press, 1994.

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