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Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication[1]. Moreover, the proposed 5G standard[2] relies on the closely related polar codes[3] for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science.

Reed-Muller code RM(r,m)
Named afterIrving S. Reed and David E. Muller
Classification
TypeLinear block code
Block length
Message length
Rate
Distance
Alphabet size
Notation-code

Reed–Muller codes generalize the Reed–Solomon codes and the Walsh–Hadamard code. Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs.

Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ rm, the Reed–Muller code with parameters r and m is denoted as RM(rm). When asked to encode a message consisting of k bits, where to produce a holds, the RM(rm) code produces a codeword consisting of 2m bits.

Reed–Muller codes are named after David E. Muller, who discovered the codes in 1954[4], and Irving S. Reed, who proposed the first efficient decoding algorithm[5].

Contents

Description using low-degree polynomialsEdit

Reed–Muller codes can be described in several different (but ultimately equivalent) ways. The description that is based on low-degree polynomials is quite elegant and particularly suited for their application as locally testable codes and locally decodable codes.[6]

EncoderEdit

A block code can have one or more encoding functions   that map messages   to codewords  . The Reed–Muller code RM(r, m) has message length   and block length  . One way to define an encoding for this code is based on the evaluation of multilinear polynomials with m variables and total degree r. Every multilinear polynomial over the finite field with two elements can be written as follows:

 
The   are the variables of the polynomial, and the values   are the coefficients of the polynomial. Since there are exactly   coefficients, the message   can be used as these coefficients. In this way, each message   gives rise to a unique polynomial   in m variables. To construct the codeword  , the encoder evaluates   at all evaluation points  , where it interprets the sum as addition modulo two in order to obtain a bit  . That is, the encoding function is defined via
 

The fact that the codeword   suffices to uniquely reconstruct   follows from Lagrange interpolation, which states that the coefficients of a polynomial are uniquely determined when sufficiently many evaluation points are given. Since   and   holds for all messages  , the function   is a linear map. Thus the Reed–Muller code is a linear code.

ExampleEdit

For the code RM(2, 4), the parameters are as follows:

 

Let   be the encoding function just defined. To encode the string x = 1 1010 010101 of length 11, the encoder first constructs the polynomial   in 4 variables:

 
Then it evaluates this polynomial at all 16 evaluation points:
 
As a result, C(1 1010 010101) = 1111 1010 0101 0000 holds.

DecoderEdit

As was already mentioned, Lagrange interpolation can be used to efficiently retrieve the message from a codeword. However, a decoder needs to work even if the codeword has been corrupted in a few positions, that is, when the received word is different from any codeword. In this case, a local decoding procedure can help.

Generalization to larger alphabets via low-degree polynomialsEdit

Using low-degree polynomials over a finite field   of size  , it is possible to extend the definition of Reed–Muller codes to alphabets of size  . Let   and   be positive integers, where   should be thought of as larger than  . To encode a message   of with  , the message is again interpreted an  -variate polynomial   of total degree at most   and with coefficient from  . Such a polynomial indeed has   coefficients. The Reed–Muller encoding of   is the list of all evaluations of   over all  . Thus the block length is  .

Description using a generator matrixEdit

A generator matrix for a Reed–Muller code RM(r, m) of length N = 2m can be constructed as follows. Let us write the set of all m-dimensional binary vectors as:

 

We define in N-dimensional space   the indicator vectors

 

on subsets   by:

 

together with, also in  , the binary operation

 

referred to as the wedge product (not to be confused with the wedge product defined in exterior algebra). Here,   and   are points in   (N-dimensional binary vectors), and the operation   is the usual multiplication in the field  .

  is an m-dimensional vector space over the field  , so it is possible to write

 

We define in N-dimensional space   the following vectors with length   and

 

where 1 ≤ i ≤ m and the Hi are hyperplanes in   (with dimension m − 1):

 

The generator matrixEdit

The Reed–Muller RM(r, m) code of order r and length N = 2m is the code generated by v0 and the wedge products of up to r of the vi, 1 ≤ im (where by convention a wedge product of fewer than one vector is the identity for the operation). In other words, we can build a generator matrix for the RM(r, m) code, using vectors and their wedge product permutations up to r at a time  , as the rows of the generator matrix, where 1 ≤ ikm.

Example 1Edit

Let m = 3. Then N = 8, and

 

and

 

The RM(1,3) code is generated by the set

 

or more explicitly by the rows of the matrix:

 

Example 2Edit

The RM(2,3) code is generated by the set:

 

or more explicitly by the rows of the matrix:

 

PropertiesEdit

The following properties hold:

  1. The set of all possible wedge products of up to m of the vi form a basis for  .
  2. The RM (r, m) code has rank
     
  3. RM (r, m) = RM (r, m − 1) | RM (r − 1, m − 1) where '|' denotes the bar product of two codes.
  4. RM (r, m) has minimum Hamming weight 2mr.

ProofEdit

  1. There are
     

    such vectors and   have dimension N so it is sufficient to check that the N vectors span; equivalently it is sufficient to check that  .

    Let x be a binary vector of length m, an element of X. Let (x)i denote the ith element of x. Define

     

    where 1 ≤ im.

    Then  

    Expansion via the distributivity of the wedge product gives  . Then since the vectors   span   we have  .
  2. By 1, all such wedge products must be linearly independent, so the rank of RM(r, m) must simply be the number of such vectors.
  3. Omitted.
  4. By induction.
    The RM(0, m) code is the repetition code of length N =2m and weight N = 2m−0 = 2mr. By 1   and has weight 1 = 20 = 2mr.
    The article bar product (coding theory) gives a proof that the weight of the bar product of two codes C1 , C2 is given by
     
    If 0 < r < m and if
    1. RM(r,m − 1) has weight 2m−1−r
    2. RM(r − 1,m − 1) has weight 2m−1−(r−1) = 2mr
    then the bar product has weight
     

Decoding RM codesEdit

RM(r, m) codes can be decoded using majority logic decoding. The basic idea of majority logic decoding is to build several checksums for each received code word element. Since each of the different checksums must all have the same value (i.e. the value of the message word element weight), we can use a majority logic decoding to decipher the value of the message word element. Once each order of the polynomial is decoded, the received word is modified accordingly by removing the corresponding codewords weighted by the decoded message contributions, up to the present stage. So for a rth order RM code, we have to decode iteratively r+1, times before we arrive at the final received code-word. Also, the values of the message bits are calculated through this scheme; finally we can calculate the codeword by multiplying the message word (just decoded) with the generator matrix.

One clue if the decoding succeeded, is to have an all-zero modified received word, at the end of (r + 1)-stage decoding through the majority logic decoding. This technique was proposed by Irving S. Reed, and is more general when applied to other finite geometry codes.

Description using a recursive constructionEdit

A Reed–Muller code RM(r,m) exists for any integers   and  . RM(m, m) is defined as the universe ( ) code. RM(−1,m) is defined as the trivial code ( ). The remaining RM codes may be constructed from these elementary codes using the length-doubling construction

 

From this construction, RM(r,m) is a binary linear block code (n, k, d) with length n = 2m, dimension   and minimum distance   for  . The dual code to RM(r,m) is RM(m-r-1,m). This shows that repetition and SPC codes are duals, biorthogonal and extended Hamming codes are duals and that codes with k = n/2 are self-dual.

Special cases of Reed–Muller codesEdit

Table of all RM(r,m) codes for m≤5Edit

All RM(rm) codes with   and alphabet size 2 are displayed here, annotated with the standard [n,k,d] coding theory notation for block codes. The code RM(rm) is a  -code, that is, it is a linear code over a binary alphabet, has block length  , message length (or dimension) k, and minimum distance  .

RM(m,m)
(2m, 2m, 1)
universe codes
RM(5,5)
(32,32,1)
RM(4,4)
(16,16,1)
RM(m − 1, m)
(2m, 2m−1, 2)
SPC codes
RM(3,3)
(8,8,1)
RM(4,5)
(32,31,2)
RM(2,2)
(4,4,1)
RM(3,4)
(16,15,2)
RM(m − 2, m)
(2m, 2mm−1, 4)
ext. Hamming codes
RM(1,1)
(2,2,1)
RM(2,3)
(8,7,2)
RM(3,5)
(32,26,4)
RM(0,0)
(1,1,1)
RM(1,2)
(4,3,2)
RM(2,4)
(16,11,4)
RM(0,1)
(2,1,2)
RM(1,3)
(8,4,4)
RM(2,5)
(32,16,8)
self-dual codes
RM(−1,0)
(1,0, )
RM(0,2)
(4,1,4)
RM(1,4)
(16,5,8)
RM(−1,1)
(2,0, )
RM(0,3)
(8,1,8)
RM(1,5)
(32,6,16)
RM(−1,2)
(4,0, )
RM(0,4)
(16,1,16)
RM(1,m)
(2m, m+1, 2m−1)
Punctured hadamard codes
RM(−1,3)
(8,0, )
RM(0,5)
(32,1,32)
RM(−1,4)
(16,0, )
RM(0,m)
(2m, 1, 2m)
repetition codes
RM(−1,5)
(32,0, )
RM(−1,m)
(2m, 0, ∞)
trivial codes

Properties of RM(r,m) codes for r≤1 or r≥m-1Edit

  • RM(0, m) codes are repetition codes of length N = 2m, rate   and minimum distance  .
  • RM(1, m) codes are parity check codes of length N = 2m, rate   and minimum distance  .
  • RM(m − 1, m) codes are single parity check codes of length N = 2m, rate   and minimum distance  .
  • RM(m − 2, m) codes are the family of extended Hamming codes of length N = 2m with minimum distance  .[7]

ReferencesEdit

  1. ^ Massey, James L. (1992), "Deep-space communications and coding: A marriage made in heaven", Advanced Methods for Satellite and Deep Space Communications, Lecture Notes in Control and Information Sciences, 182, Springer-Verlag, pp. 1–17, CiteSeerX 10.1.1.36.4265, doi:10.1007/bfb0036046, ISBN 978-3540558514[pdf]
  2. ^ "3GPP RAN1 meeting #87 final report". 3GPP. Retrieved 31 August 2017.
  3. ^ Arikan, Erdal (2009). "Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels - IEEE Journals & Magazine". IEEE Transactions on Information Theory. 55 (7): 3051–3073. doi:10.1109/TIT.2009.2021379. hdl:11693/11695. Retrieved 2018-08-31.
  4. ^ Muller, David E. (1954). "Application of Boolean algebra to switching circuit design and to error detection". Transactions of the I.R.E. Professional Group on Electronic Computers. EC-3 (3): 6–12. doi:10.1109/irepgelc.1954.6499441. ISSN 2168-1740.
  5. ^ Reed, Irving S. (1954). "A class of multiple-error-correcting codes and the decoding scheme". Transactions of the IRE Professional Group on Information Theory. 4 (4): 38–49. doi:10.1109/tit.1954.1057465. ISSN 2168-2690.
  6. ^ Prahladh Harsha et al., Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes), Section 5.2.1.
  7. ^ Trellis and Turbo Coding, C. Schlegel & L. Perez, Wiley Interscience, 2004, p149.

Further readingEdit

External linksEdit