Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order reed muller codes are described. This is a type of decoding where we take our received word as given and have each bit. In exemplary embodiments, a method comprises receiving a codeword encoded with a reedmuller code, generating a pattern to retrieve voting bits, decoding the codeword based on the voting bits and, and providing the decoded codeword. Testing was performed both in software using a simulator and in hardware using the ni usrp software defined radio. These codes were initially given as binary codes, but modern generalizations to qary codes exist. Pdf softdecision majority decoding of reedmuller codes. Using the symmetric group of rm and polar codes, the specific decoding algorithm can be carried out on multiple permutations. The answer to your question depends on your definition of reedmuller codes and what is meant by decoding. Polynomial codes over certain finite fields, journal of the society of industrial and applied mathematics, pp. Reedsolomon codes reed and solomon, 1960 are a special class of bch codes. Reedmuller rm codes are among the oldest families of errorcorrecting. Reedsolomon encoder and decoder are commonly used in data transmission and storage applications, such as broadcast equipment, wireless lans, cable. This is a research project i completed for a course at carleton university, finite fields and coding theory.
The reed solomon decoder logicore module is a high speed, compact design that implements many different reed solomon coding standards including g. In the galois array decoded, each row represents the attempt at decoding the corresponding row in code. The core is fully synchronous, using a single clock, and supports continuous input data with no gap between code blocks. Green, a serial orthogonal decoder, jpl space programs summary, vol. Abstract we consider the problem of decoding firstorder reedmuller codes ef. We give an algorithm that implicitly adapts to the noise conditions, runs signi. They were discovered by muller and provided with a decoding algorithm by reed in 1954. It should be noted that multibit decoding for rate1, rate0, and repetition constituent codes was proposed in the context of likelihoodbased reed muller rm decoders 23 and likelihoodbased. In exemplary embodiments, a method comprises receiving a codeword encoded with a reed muller code, generating a pattern to retrieve voting bits, decoding the codeword based on the voting bits and, and providing the decoded codeword. This is an improvement over the proof using onepoint algebraicgeometric codes given in \citepw2005.
A decoding failure occurs if rsdec detects more than nk2 errors in a row of code. Towards this, we will convert reed solomon codes in to binary codes. Fast decoding algorithms for first order reedmuller and related. Efficiently decoding reed muller codes from random errors authors. Listdecoding reedmuller codes over f 2 by pairwise independence, the overall errorrate on a0 is likely to be. Introduction to readmuller logic computer action team. Unique decoding local listdecoding johnson bound global listdecoding figure 1. Vhdl design and fpga implementation of a parallel reedsolomon15,k,d. This comprises a paper that attempts to present reedmuller codes in a simplistic way, and includes a python implementation of encoding and decoding using majority logic using reedmuller codes.
Reedmuller codes were defined and encoding matrices were discussed. The standard decoding algorithm for reed muller codes uses majoritylogic decoding which in turn requires harddecision demodulation. Vhdl design and fpga implementation of reed solomon encoder. The canonical definition of reed muller codes is as nonsystematic codes see e. Maximum likelihood decoding of reed solomon codes conference. This result solves the open problem of whether or not reedmuller codes achieve capacity for some rate bounded away from 0 and 1. We prove that the listdecoding radius for quadratic polynomials equals 12q over any field fq where q 2. On decoding of reedmuller codes in semicontinuous channels. Coding theory lecture notes nathan kaplan and members of the tutorial september 7, 2011. A method and apparatus for encoding and decoding reedmuller codes are provided. In this paper, we explore the question of whether the code symmetry of rm codes can also be exploited to achieve nearml performance in practice. Abstractreed muller rm codes exhibit good performance under maximumlikelihood ml decoding due to their highly symmetric structure. As a result, the presented hybrid decoding algorithm reduces the number of floatingpoint. Fast decoding algorithms for first order reedmuller and.
Reedmuller codes for random erasures and errors princeton. Mar 23, 2015 this time well define the reedsolomon code which optimizes a different bound called the singleton bound, and then generalize them to a larger class of codes called reedmuller codes. Example showing how to program gfpgfr for a generator polynomial. Reed muller codes are relatively easy to decode, and firstorder codes are especially efficient. So the generator polynomial of a reedsolomon code of designed distance. A noiseadaptive algorithm for firstorder reedmuller decoding jon feldman ibrahim aboufaycal matteo frigo abstract we consider the problem of decoding firstorder reedmuller codes ef. In future posts well consider algorithmic issues behind decoding the codes, for now we just care about their existence and optimality properties. If n is not exactly 2 m1, rsdec assumes that code is a corrupted version of a shortened code. A maximum likelihood decoding algorithm is described in section 4 and a specific example is used to illustrate the process. Reedmuller codes, hybrid decoding, errorcorrecting codes, digital communications. Image correction using reed muller code a paper by siva. Section 5 presents the aci performance, and section 6 assesses the performance of the decoder for a number of realistic channel conditions. Image correction using reed muller code a paper by.
Recursive projectionaggregation decoding of reedmuller. A compact and fast fpga based implementation of encoding and decoding algorithm using reed. We then propose a new low complexity recursive algebraic decoding algorithm for reed muller and product reed solomon codes. Some errorcorrecting codes and their applications j. Second order reedmuller decoding algorithm in quantum. A reedsolomon code is a bch code over gfq of length n q 1, that is, m 1. The algorithm repeatedly relegates itself to the shorter rm codes by. Fast decoding algorithms for first order reedmuller and related codes. A hybrid decoding of reedmuller codes sage journals. Asking for help, clarification, or responding to other answers. Proceedings of 2nd international workshop on algebraic and combinatorial coding theory, leningrad, 1990, pp. Reedsolomon encoding and decoding can be carried out in software or in specialpurpose hardware. The software supports addition of errors, simulating faults in the transmission channel. Vhdl design and fpga implementation of reed solomon.
A hybrid decoding of reedmuller codes shuang li, shicheng. Home conferences stoc proceedings stoc 15 reed muller codes for random erasures and errors. The standard decoding algorithm for reedmuller codes uses majoritylogic decoding which in turn requires harddecision demodulation. We give an algorithm that implic itly adapts to the noise conditions, runs signi. Soft and hard decisions maximum likelihood decoding algorithms for first order reedmuller and the. Decoding reedmuller codes using minimum weight parity checks abstract. In this article, a hybrid decoding algorithm for reedmuller codes is presented. Encoding and decoding can be done for better transmission of the signal without the. Reed solomon encoder and decoder rutgers university. The power saving given by reedsolomon in decibels is the coding gain. For reedmuller codes, the selection is based on the hamming weight of the row. Architectures for encoding and decoding reedsolomon codes.
Performance of reedmuller codes with ofdm and a maximum. Reedmuller rm codes were discovered by reed and muller in 1954 7, 8. This example shows how to set up the reedsolomon rs encoderdecoder to use punctured codes. The answer to your question depends on your definition of reed muller codes and what is meant by decoding. Reed solomon encoder and decoder provide a compromise between efficiency and complexity, so that this can be easily implemented using hardware or fpga.
Then we shall return to our original goal of constructing explicit codes with constant relative distance and rate. Reedsolomon encoding and decoding implementation example. The reedsolomon code has a very simple definition, especially for those of you who have read about secret sharing given a prime power and integers, the reedsolomon code with these parameters is defined by its encoding function as follows generate explicitly pick distinct elements a message is a list of elements. Reedsolomon allows the system to achieve this target ber with a lower transmitter output power. Abstract reedmullercodesencodeanmvariatepolynomialofdegreer byevaluatingitonallpoints in 0,1m.
Reedmuller rm codes exhibit good performance under maximumlikelihood ml decoding due to their highlysymmetric structure. Us20090031195a1 method and apparatus for encoding and. Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order reedmuller codes are described. We will restrict our investigation to the binary case. Software shows the individual steps of encodingdecoding. Reed solomon encoder and decoder falls in the category of forward error correction encoders and it is optimized for burst errors rather than bit errors. Reedmuller rm and polar codes are a class of capacityachieving channel coding schemes with the same factor graph representation.
Softdecision majority decoding of reedmuller codes article pdf available in ieee transactions on information theory 461. Software for the st171 for serial numbers that start with 09 or. The decoding process was given and some examples were demonstrated to clarify the. Unlike the conventional algorithm, the presented algorithm ends recursive decomposition when r1,m and rm. This comprises a paper that attempts to present reed muller codes in a simplistic way, and includes a python implementation of encoding and decoding using majority logic using reed muller codes. Home conferences stoc proceedings stoc 15 reedmuller codes for random erasures and errors.
Decoding reedmuller codes using minimum weight parity. We present a fourieranalytic approach to list decoding reed muller codes over arbitrary finite fields. A noiseadaptive algorithm for firstorder reedmuller. Rm codes have been extensively studied with respect to decoding errors in both the worst case and random setting. Decoding reedmuller and polar codes by successive factor. Software implementation of the reedsolomon encoder and decoder, and additionally parts of the baseband transmitter and receiver were done in labview. Ramprasad saptharishi, amir shpilka, ben lee volk submitted on 31 mar 2015 v1, last revised 27 aug 2015 this version, v2.
Program in applied and computational mathematics, and department of electrical. Reedmuller codes reedmuller codes are among the oldest known codes and have found widespread applications. Channel coding theory reedmuller codes these codes were discovered by muller and the decoding by reed in 1954. It should be noted that multibit decoding for rate1, rate0, and repetition constituent codes was proposed in the context of likelihoodbased reedmuller rm decoders 23. A noiseadaptive algorithm for firstorder reedmuller decoding. Reedmuller codes mathematical and statistical sciences. Recursive list decoding is considered for reedmuller rm codes. A method and apparatus for encoding and decoding reed muller codes are provided. Program through the national research foundation of. This was a posted open problem from the 2015 simons institute program on information theory and the subject of. This confirms a conjecture due to gopalan, klivans and.
Soft and hard decisions maximum likelihood decoding algorithms for first order reed muller and the nordstromrobinson codes with low. For this reason, i posted a long answer to the question. Decoding reedmuller codes using minimumweight parity checks. The canonical definition of reedmuller codes is as nonsystematic codes see e. In 1972, a reed muller code was used by mariner 9 to transmit black and white photographs of mars. This is a type of decoding where we take our received word as given and have each bit vote on what the message sent was. The objective of bachelor thesis is aimed to develop an encoderdecoder of selected reedmuller codes. Implementation of encoding and decoding algorithm using reed solomon codes. Reedmuller codes are among the oldest known codes and have found widespread applications. Muller decoder, we derive a method to decode the second order. The reedsolomon decoder logicore module is a high speed, compact design that implements many different reedsolomon coding standards including g. This work investigates the properties of the index sets pointing to those rows in the infinite blocklength limit.
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