Closest coset decoding of mod u mod u+v mod codes

Hemmati, F.

doi:10.1109/49.29620

Cited by 37 publications

(19 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For any received symbol , one can readily calculate the two posterior probabilities of sending a and a , respectively. Then we find their (3) Given that the symbol is being transmitted over the channel , the first two moments of the random variable (RV) equal (4) We shall see that our decoding thresholds can be defined in terms of parameter (5) which is defined by the pdf . The main result is given in the following theorem.…”

Section: Summary Of the Resultsmentioning

confidence: 99%

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Dumer

2006

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

Abstract-Soft-decision decoding is considered for general Reed-Muller (RM) codes of length and distance used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order ln and corrects most error patterns of the Euclidean weight of order ln instead of the decoding threshold 2 of the bounded distance decoding. Also, for long RM codes of fixed rate , the new algorithm increases 4 times the decoding threshold of its hard-decision counterpart.

show abstract

Section: Summary Of the Resultsmentioning

confidence: 99%

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Dumer

2006

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

show abstract

“…Renewed interest in lulu+vl constructions, and RM codes in particular, seems to have been awakened as part of a wave of interrelated ideas known as "multi-level coding" [17], "generalized concatenated coding" [18], [19], and "trellis based representations and decoding of block codes" [20]. Examples of papers that treat RM codes from these broader perspectives are [9], [21], and [22]. An exposition of this line of work on recursive decoders designed for multi-level lulu + vi codes, and RM codes in particular, is given in [4,Chapter 15].…”

Section: Recursive Decodersmentioning

confidence: 99%

A survey of reed-muller codes from polar coding perspective

Arıkan

2010

IEEE Information Theory Workshop 2010 (ITW 2010)

View full text Add to dashboard Cite

A survey of Reed-Muller (RM) coding is given with the goal of establishing a continuity between RM codes and polar codes. The focus is mainly on recursive decoding methods for RM codes and other ideas that are most relevant to polar coding. Index Terms-Reed-Muller codes, polar codes, channel polar ization. I. RM CODES RM codes were discovered by Muller [1] and Reed [2]. For every pair of integers 0 :::; r :::; m, there is an RM code, denoted RM(r, m), with block-length n = 2m, dimension k = L�= o (7), and minimum distance d = 2m-T• For each m 2: 0, the code RM(O, m) is the repetition code and RM(m, m) is the trivial code consisting of all binary n-tuples. All other RM codes can be obtained by the Plotkin construction [3], also known as the lulu + v i construction, by the recursion(1)A more explicit algebraic description of RM codes is given by first specifying a generator matrix for RM(m, m) aswhere F ® m denotes the mth Kronecker power of F. Then, for o :::; r :::; m, GRM(r, m), the generator matrix of RM(r, m), is taken as the submatrix of G RM (m, m) consisting of rows with Hamming weights 2: 2m-T• The equivalence of these two constructions is well-known (see, e.g., [4, p. 114]) .For example, for m = 3, we have 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 GRM(3,3) = 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 (3) 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 and RM(l, 3) is the code with generator matrix G RM(I, 3) � [ l 1 1 1 0 0 0 ] 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1The encoding operation for all RM codes of a given length n = 2m can be represented universally as a transform x= dF ® m (5) where d = (do, ... , dn-1 ) and x = (xo, ... , Xn-l ) represent data and code words, respectively. Depending on the order of the RM code to be encoded, some of the elements of dare fixed to zero. For example, for the RM(l, 3) code, the fixed coordinates are do, d1, d2, and d4. The coordinates that are not fixed are free to take on any 0-1 value.The universal encoder can be implemented by a circuit of complexity O(n log n), as shown in Fig. 1 for n = 8. The figure also shows how a codeword x splits into two parts u and v, each of which in tum splits again into two parts, and so on, until one reaches single data elements di. For example, u has the form Irlr+sl, and s has the form Id5Id4+d51. Thus, this code has a three-level lulu + v i structure. U o Y 4 Fig. I. Encoding and transmission of data.A codeword x is obtained by a circuit that mUltiplies a data word d by F0a. The codeword is sent over a memory less channel Wand a channel output y is received.The transform (5) can be used as an encoder for a broader class of codes than just RM codes. For a given pair of code parameters (n, k), there are (�) codes that can be encoded, each corresponding to a particular way of choosing the (n -k)

show abstract

“…[4][5][6][7]9 We propose a new recursive decoding algorithm for the RM code. Here, the recursive decomposition processes are ended one step earlier than the conventional one.…”

Section: A Hybrid Recursive Decoding For Rm Codesmentioning

confidence: 99%

A hybrid decoding of Reed–Muller codes

Zhang

Chen

et al. 2017

International Journal of Distributed Sensor Networks

View full text Add to dashboard Cite

In this article, a hybrid decoding algorithm for Reed-Muller codes is presented. Unlike the conventional algorithm, the presented algorithm ends recursive decomposition when R(1, m) and R(m À 1, m) appeared. A simplified maximumlikelihood algorithm based on fast Hadamard transform is also exploited to decode the systematic code through its special structure. As a result, the presented hybrid decoding algorithm reduces the number of floating-point multiplications significantly as compared with the conventional algorithms. In addition, the new algorithm has better error performance than the conventional ones.

show abstract

Closest coset decoding of mod u mod u+v mod codes

Cited by 37 publications

References 2 publications

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

A survey of reed-muller codes from polar coding perspective

A hybrid decoding of Reed–Muller codes

Contact Info

Product

Resources

About