Decoding reedmuller codes using minimum weight parity. You could use the hadamardn command to make a hadamard matrix. They were discovered by muller and provided with a decoding algorithm by reed in 1954. Jointly optimized multiple reedmuller codes for wireless. This is a research project i completed for a course at carleton university, finite fields and coding theory. However, this performance is obtained with maximumlikelihood decoding which can be costly for practical applications. These codes were initially given as binary codes, but modern generalizations to qary codes exist. Mahanti, submitted to am, pdf file efficient deterministic compressed sensing for images with chirps and reedmuller codes with k. Eurasip journal on wireless communications and networking jointly optimized multiple reedmuller codes for wireless halfduplex codedcooperative network with joint decoding saqib ejaz 0 3 fengfan yang 0 3 hongjun xu 2 shunwai zhang 1 0 college of electronic and information engineering, nanjing university of aeronautics and astronautics, nanjing. The weight enumerators for certain subcodes of the second order binary reedmuller codes by e.
We define a johnson graph code as a subspace of labelings of the vertices in a johnson graph with the property that labelings are uniquely determined by their restriction to vertex. Pdf this paper presents a set of quantum reedmuller codes which are typically 100 times more effective than existing quantum reedmuller codes. Coding theory lecture notes nathan kaplan and members of the tutorial september 7, 2011 these are the notes for the 2011 summer tutorial on coding theory. Reedmuller rm codes are classical codes that have enjoyed unabated interest since their introduction in 1954 due to their simple recursive structure and lowcomplexity decoding. Take out the vectors which form the basis for this matrix this is ther generator of the hadamard code then products of the rows in the generator will generate any reedmuller code.
The standard decoding algorithm for reedmuller codes uses majoritylogic decoding which in turn requires harddecision demodulation. These codes are important from points of view of coding theory, design theory and. And similarly, u3 was this matrix that weve now seen many times. The main goal of the thesis is to develop new coding techniques that. Reedmuller codes, basis of minimum weight codewords, incidence matrix, standard form mr subject classi. In proceedings of the 40th annual acm symposium on theory of computing, victoria, british columbia, canada, may 1720, 2008, pages 265274, 2008. These results also show that short algebraic codes such as reedmuller rm and extended bch ebch codes under ml decoding tie or outperform all the other choices.
I have not gone through and given citations or references for all of the results given here, but the presentation relies heavily on two sources, van. Efficiently listdecodable punctured reedmuller codes authors. This paper presents a set of quantum reedmuller codes which are typically 100 times more effective than existing quantum reedmuller codes. And you see how all the reedmuller codes are generated. In this paper, we consider the reedmuller rm codes. Softdecision decoding is considered for general reedmuller rm codes of length n and distance d used over a memoryless channel. A nonzero codeword c of c is called minimal if its support does not contain the support of any other nonzero codeword of c as a proper subset. Reedmuller codes reedmuller codes are among the oldest known codes and have found widespread applications. In this paper, we give a sufficient and necessary condition for the covering radius of rm3, 7 to be equal to 22. Introduction to the theory of errorcorrecting codes, third edition is the ideal textbook for seniorundergraduate and firstyear graduate courses on errorcorrecting codes in mathematics. Figure 1 reedsolomon code definitions also, a reedsolomon code is a linear code adding two code words produces another code word and it is cyclic cyclically shifting the symbols of a code word produces another code word. A recursive decoding algorithm is designed and its decoding threshold is derived for long rm codes. Codes of the latter type are in general different from affine reedmuller codes, but for the special case of the hypercube they agree with binary reedmuller codes.
We can find generators for all the reedmuller codes from this set. Minimum distance for m5 and r 2 then n 32, k 16,and dmin 8. Reedmuller codes associated to projective algebraic varieties. Designs in codes, perfect codes, selforthogonal codes, doubly even codes selfdual codes, golay codes, assmusmattson theorem, extremal codes. Efficiently listdecodable punctured reedmuller codes.
It belongs to the family of bosechaudhurihocquenghem bch codes 3, 4, but is distinguished by. Reedsolomon, bch, reedmuller, and concatenated codes february 2010 lecturer. Venkatesan guruswami, lingfei jin, chaoping xing submitted on 3 aug 2015 v1, last revised 2 apr 2017 this version, v3. The reason theyre so useful is not necessarily because they optimize some bound if they do, i havent heard of it, but because they specialize to all sorts of useful codes with useful properties. The family of reedmuller codes also includes polynomial evaluation codes where the total degree d is larger than jsj, and the individual degree is capped to be at most jsj 1. And its probability density function pdf tends to the gaussian. The covering radius of the third order reedmuller code of length 128 has been an open problem for many years. Local unambiguous decoding of some hadamard codes and reedmuller codes. We studyminimal codewords of weight smaller than 32m. The message space of the rthorder reedmuller code rmr,m consists of degree r polynomials over f 2 in m variables. Using this condition, we prove that the covering radius of rm3, 7 in rm4, 7 is 20. We will restrict our investigation to the binary case.
Weight enumerators, macwilliams identity, codes in designs, projective plane of order 10 stirling numbers of the 2 nd kind, pless power moments, gleason polynomials. On the covering radius of the third order reedmuller code. If v is of degree h, we give parameters of reed muller codes of order d probability density function a general setting of codes on varieties which comprises both the one point geometric goppa codes as the qary reedmuller codes. Some errorcorrecting codes and their applications j. For the first order rm code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order rm code. Papers and preprints on image and signal analysis, medical imaging stability of efficient deterministic compressed sensing for images with chirps and reedmuller sequences with k.
The description that is based on lowdegree polynomials is quite elegant and particularly suited for their application as locally testable codes and locally decodable codes encoder. They are used in particular secret sharing schemes, to model the access structures. The reedmuller code is a neat generalization of the reedsolomon code to multivariable polynomials. A block code can have one or more encoding functions. For this reason, i posted a long answer to the question. 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. For any m and r, 0 r m, the binary rth order reedmuller code. Local testing of reedmuller codes from this chapter onwards, we will switch gears and talk about property testing of codes. This website uses cookies to ensure you get the best experience on our website. The encoding of a polynomial p is the vector of its evaluations at all points in fm 2. Thats what we called our universal reedmuller generator matrix for m equals 2. Thus rmr,m is an n,k,d 2 errorcorrecting code where n 2m,k p i. Minimal codewords in reedmuller codes ghent university. David eugene muller november 2, 1924 april 27, 2008 was an american mathematician and computer scientist.
Thas january, 2009 abstract minimal codewords were introduced by massey 8 for cryptographical purposes. Enhanced recursive reedmuller erasure decoding arxiv. When m 1, this code is known as the reedsolomon code 18, and for m 1 this code is a reed muller code 14, 17. Reedmuller codes can be described in several different but ultimately equivalent ways. Hence, although shannons limit seems finally close at hand, new challenges are just around the corner.
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