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Volume 3, Issue 8, August 2013 ISSN: 2277 128X
International Journal of Advanced Research in
Computer Science and Software Engineering
Research Paper
Available online at: www.ijarcsse.com
Error Detection and Correction Using Reed Solomon Codes
Priyanka Shrivastava1 Uday Pratap Singh2
M.Tech Scholar Computer Science Professor Computer Science
Department L.N.C.T, Bhopal, India Department L.N.C.T, Bhopal, India
Abstract-- In the present world, communication system which includes wireless, satellite and space communication,
reducing error is being critical. During message transferring the data might get corrupted, so high bit error rate of the
wireless communication system requires employing to various coding methods for transferring the data. Channel
coding for detection and correction of error helps the communication systems design to reduce the noise effect during
transmission. The purpose of this paper is to study and analyze the performance and efficiency of Reed-Solomon (RS)
Codes. In coding theory, Reed-Solomon (RS) codes are the subset of BCH codes that are one of the most powerful
known classes of linear cyclic block codes. Reed-Solomon (RS) codes are very efficient and best for correction of burst
errors and have a wide range of applications in digital communication and data storage. Reed-Solomon (RS) is the
most powerful technique used for error detection and correction at present.
Keyword- Reed-Solomon (RS), Galois Field (GS), Generator Polynomial g(x), Block length, Bit Error Rate (BER),
Signal Noise Ratio (SNR)
I. Introduction
Channel coding is an important signal processing operation for the efficient transmission of digital information over the
channel. In channel coding the number of symbols in the source encoded message is increased in a controlled manner in
order to facilitate two basic objectives at the receiver one is Error detection and other is error correction. Error detection
and error correction to achieve good communication is also employed in devices. It is used to reduce the level of noise
and interferences in electronic medium. The amount of error detection and correction required and its effectiveness
depends on the signal to noise ratio (SNR) [1]. A channel code is a broadly used term mostly referring to the forward
error correction code. Forward error correction (FEC) is a system of error control for data transmission, whereby the
sender adds redundant data to its messages, also known as an error correction code. This allows the receiver to detect and
correct errors without the need to ask the sender for additional data. FEC is applied where retransmissions are relatively
costly or impossible. FEC information is usually added to most mass storage devices to protect against damage to the
stored data [2]. There are many types of block codes, but the most notable is Reed Solomon coding, Golay, BCH,
Multidimensional parity, and Hamming codes are other example of block codes.
Reed Solomon is an error-correcting coding system that was devised to address the issue of correcting multiple
errors – especially burst-type errors in mass storage devices (hard disk drives, DVD, barcode tags), wireless and mobile
communications units, satellite links, digital TV, digital video broadcasting (DVB), and modem technologies like xDSL
[3]. Reed-Solomon codes are an important subset of non-binary cyclic error correcting code and are the most widely used
codes in practice. These codes are used in wide range of applications in digital communications and data storage. Reed
Solomon describes a systematic way of building codes that could detect and correct multiple random symbol errors. By
adding t check symbols to the data, an RS code can detect any combination of up to t erroneous symbols, or correct up to
⌊t/2⌋ symbols. Furthermore, RS codes are suitable as multiple-burst bit-error correcting codes, since a sequence of b + 1
consecutive bit errors can affect at most two symbols of size b. The choice of t is up to the designer of the code, and may
be selected within wide limits.
Example:
A popular Reed Solomon code is RS (255,223) with 8-bit symbols. Each codeword contains 255 code word bytes, of
which 223 bytes are data and 32 bytes are parity for this code
n = 255, k = 223, s = 8
2t = 32, t = 16
The decoder can correct any 16 symbol errors in the code word: i.e. errors in up to 16 bytes anywhere in the codeword
can be automatically corrected.
Given a symbol size s, the maximum codeword length (n) for a Reed Solomon code is n = 2s – 1.
For example, the maximum length of a code with 8-bit symbols (s=8) is 255 bytes.
Reed Solomon codes may be shortened by (conceptually) making a number of data symbols zero at
the encoder, not transmitting them, and then re-inserting them at the decoder.
RS are block codes and are represented as RS (n, k), where n is the size of code word length and k is the number data
symbols, n – k = 2t is the number of parity symbols as shown in Fig. 1.
Shrivastava et al., International Journal of Advanced Research in Computer Science and Software Engineering 3(8),
August - 2013, pp. 965-969
© 2013, IJARCSSE All Rights Reserved Page | 966
k 2t
n
Fig. 1 Structure of RS codeword
Each symbol at input is composed of m bits given by relation n = qm-1. The message is divided into k symbols and 2t
parity symbols are added to give n symbols of RS code. Reed Solomon codes are constructed using generator polynomial
which is given by equation:
g (x) = (x - αi) (x - αi+1)……( x - α2t+i-1) ( x - α2t+i)
RS encoding is based on addition operation and multiplication operation over Galois field (GF(2m)). For GF(2m) field m
XOR gates will be required for the realization of addition operation and operation of multiplication requires m2 AND
gates & m2-1 XOR gates using conventional approaches [4].
II. PROPERTIES OF REED SOLOMON CODES
The properties of Reed-Solomon codes make them especially suited to the applications where burst error occurs. This is
because:-
It does not matter to the code how many bits in a symbol are incorrect, if multiple bits in a symbol are corrupted
it only counts as a single error. Alternatively, if a data stream is not characterized by error bursts or drop-outs
but by random single bit errors, a Reed-Solomon code is usually a poor choice. More effective codes are
available for this case.
Designers are not required to use the natural sizes of Reed-Solomon code blocks. A technique known as
"shortening" produces a smaller code of any desired size from a larger code. For example, the widely used
(255,251) code can be converted to a (160,128). At the decoder, the same portion of the block is loaded locally
with binary zeroes [5].
A Reed–Solomon code operating on 8-bits symbols has n = 28 – 1 = 255 symbols per block because the number
of symbol in the encoded block is n = 2m – 1.
For the designer its capability to correct both burst errors makes it the best choice to use as the encoding and
decoding tool. III. REED-SOLOMON ENCODER
The Reed Solomon encoder reads in k data symbols computes the n – k symbols, append the parity symbols to the k data
symbols for a total of n symbols. The encoder is essentially a 2t tap shift register where each register is m bits wide. The
multiplier coefficients are the coefficients of the RS generator polynomial. The general idea is the construction of a
polynomial, the coefficient produced will be symbols such that the generator polynomial will exactly divide the
data/parity polynomial. From the architectural point of view, the encoder represents the set of shift registers, joined by
means of integrators and multipliers, operating according to the rules of Galois arithmetic. The shift register represents
the sequence of memory cells, called bits, each of which contains one element of a Galois field GF(q). The symbol,
contained in a specific position, is transmitted to the output line as it leaves this position. Simultaneously, the symbol
from the input line is loaded into position. Replacement of symbols takes place discretely, at strictly defined time
intervals, known as clocks. In hardware implementation of the shift register, its elements can be connected both
sequential and in parallel manner. In sequential connection, the sending of a single m-bit symbol requires m clocks, while
parallel connection requires only one clock [6].
The generator polynomial of the RS encoder is represented by
g(x) = g0 + g0 x + g0 x2 + …. + g2t-1 x2t-1 + x2t
The hardware implementation of this RS encoder is shown in Fig. 2.
Fig.2 Block Diagram of RS Encoder
DATA PARITY
Shrivastava et al., International Journal of Advanced Research in Computer Science and Software Engineering 3(8),
August - 2013, pp. 965-969
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Reed Solomon codes are based on a special area of mathematics known as Galois fields or finite fields. Finite field has
the property that arithmetic operations (+, -, x, / etc.) on field elements always have a result inside the field.
The basic principle of encoding is to find the remainder for the message divided by a generator polynomial G(x). The
encoder working is by simulating a linear feedback shift register with degree as G(x) is having, and feedback taps with
the coefficients of the generating polynomial of the cods [7].
IV. REED-SOLOMON DECODER
The Reed Solomon decoder tries to correct errors and/or erasures by calculating the syndromes for each codeword. Based
upon the syndromes the decoder is able to determine the number of errors in the received block. If there are errors
present, the decoder tries to find the locations of the errors using the Berlekamp-massey algorithm by creating an error
locator polynomial. The roots of this polynomial are found using the Chien search algorithm.
Using Forney's algorithm, the symbol error values are found and corrected. For an RS (n, k) code where n - k = 2T, the
decoder can correct up to T symbol errors in the code word. Given that errors may only be corrected in units of single
symbols (typically 8 data bits), Reed Solomon coders work best for correcting burst errors.
Decoding of Reed-Solomon codes is a complex problem that results in a bulky and extremely complicated code which
requires that the developer should have an extensive knowledge in many areas of higher mathematics. A typical decoding
is known as auto-regressive spectral decoding method, with following steps:-
1. Determining error syndrome (syndrome generator).
2. Building an error polynomial, carried out by using Barlekamp algorithms which are hard to implement or use
some simple algorithm like Euclid’s algorithm.
3. Finding the roots of this polynomial, this is usually carried out by Chein search algorithm.
4. Determining the error type, this is calculated by Forney s algorithm or any other algorithm of matrix inversion.
5. Correcting erroneous symbols by means of superimposing the mask and data word and the sequentially
inverting all bits that are corrupted via XOR operation as shown in Fig. 3 [6].
Fig. 3 Block diagram of RS Decoder
V. APPLICATIONS OF REED-SOLOMON CODES
5.1 Data Storage: -
Reed-Solomon is very widely used in mass storage systems to correct burst errors associated with media defects.
Special properties of Reed-Solomon codes make the sound quality of the CD as impressive as it is.
Reed-Solomon is a main component of CD Compact Disc. In CD a scheme known as Cross-Interleaved Reed
Solomon Coding (CIRC) is used. The result of CIRC is that it can correct up to 4000 bits error bursts, or about 2.5
mm on the surface of the disc [8].
5.2 Bar Code: -
All the two dimensional bar codes such as PDF-417, Maxi Code, Data matrix, QR Code, and Aztec Code use Reed-
Solomon error correction to allow the correct reading even if some portions of bar code are damaged. When the bar
code symbol is not recognized by the bar code scanner it will treat it as an erasure. Reed-Solomon is less common in
one-dimensional bar codes [5].
5.3 Satellite Broadcasting: -
The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television
including new channels High Definition TV and IP data. BPSK coupled with traditional Reed-Solomon and Viterbi
codes have been used for nearly 20 years for delivery of digital satellite TV [9].
Shrivastava et al., International Journal of Advanced Research in Computer Science and Software Engineering 3(8),
August - 2013, pp. 965-969
© 2013, IJARCSSE All Rights Reserved Page | 968
5.4 Spread-Spectrum Systems: -
Reed-Solomon codes can be used in designing the hopping sequences. If these sequences are carefully selected, the
interference caused by the other users in a multiple access environment can be greatly reduced [10].
5.5 Error Control for Systems with Feedback: -
Wicker and Bartz examined various means for using Reed-Solomon codes in applications that allow transmission of
information from receiver to the transmitter. These applications include mobile data transmission systems and high
reliability military communication systems [11].
5.6 Ultra Wideband (UWB): -
UWB is a wireless technology for transmitting the digital data at very high rates over a wide spectrum of frequency
by using very low power. It makes it possible to transmit data at rate over 100Mbps within 10 meters. To preserve
the important header information, MB-OFDM UWB adopts Reed-Solomon (23, 17) code. In receiver, RS decoder
needs high speed and low latency and for this efficient hardware is used [12].
VI. REED-SOLOMON ERROR PROBABILITY
Reed-Solomon codes are mainly used in correcting the burst errors. However it has its own error correcting capability.
So, error probability is useful in saving our time for detecting and correcting the error. Let us assume an example that the
code can correct 4 error symbols in an (255, 251) RS code. A maximum of 32 bits error can be corrected. So, during
decoding if the decoder calculates more than 32 bits of error while performing syndrome calculation part, then send a
signal that decoder cannot correct this error. Therefore plotting of the bit error probability (P) against the SNR will help.
There can be a range of SNR for error to be corrected.
However range includes many parts like percentage of probability that the signal will detect. Fig. 4 shows plot between
SNR and bit error probability. The code is for random 255 symbols where each symbol consists of 8 bits to be
transmitted. These 255 symbols form a code word and there are 500 such codewords. However the range estimation can
be calculated for different capability of correcting errors [10].
Fig. 4 Graph between BER and SNR
Analysis of the error probability graph, 255 symbols corresponds to m = 8, therefore symbol with contain 8 bits each and
for different range of deduced SNR different error correcting capabilities. So different number of error bits can be found
out from the plotted graph for different error capabilities. For example for 2t = 4; t = 2 therefore 16 error bits can be
corrected at max. So a range of SNR can be found out using the plotted graph. Similarly for 2t = 8; t = 4 so at max 32 bits
can be corrected. This range is quite bigger the previous range of SNR. This analysis helps us to find out that the decoder
can correct the received signal or not and this saves a lot of time and efforts.
VII. CONCLUSION
This paper presents clear understanding of Reed-Solomon codes used in error detection and correction. RS are very
powerful non-binary cyclic codes and are used mainly for burst errors. There are different applications of REED-
Solomon codes such as data storage, satellite transmission, bar code etc. and the main component of a Compact Disc is
RS code. The main purpose of this paper was to study the Reed-Solomon code encoding and decoding process and also
Shrivastava et al., International Journal of Advanced Research in Computer Science and Software Engineering 3(8),
August - 2013, pp. 965-969
© 2013, IJARCSSE All Rights Reserved Page | 969
the error probability for the RS code. The encoding process and the block diagram have been discussed and also the
different step for decoding process has been discussed. The error probability for RS code shows that the BER
performance also improves for large block length and shows a poor BER performance for lower SNR as the SNR value
increases the curve becomes steeper.
Refrences
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[5] www.wikipedia.org
[6] flylib.com/books/en/1.443.1.19/1/ a document on Reed-Solomon
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Decoder”, IEEE Trans. Comput., vol. C-34, no. 5, pp 393-403, May 1985.
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