View source for Reed–Muller code

{{Short description|Error-correcting codes used in wireless communication}}
{{infobox code
| name           = Reed-Muller code RM(r,m)
| image          =
| image_caption  =
| namesake       = [[Irving S. Reed]] and [[David E. Muller]]
| type           = [[Linear block code]]
| block_length   = <math>2^m</math>
| message_length = <math>k=\sum_{i=0}^r \binom{m}{i}</math>
| rate           = <math>k/2^m</math>
| distance       = <math>2^{m-r}</math>
| alphabet_size  = <math>2</math>
| notation       = <math>[2^m,k,2^{m-r}]_2</math>-code
}}

'''Reed–Muller codes''' are [[error-correcting code]]s that are used in wireless communications applications, particularly in deep-space communication.<ref>{{Citation|last=Massey|first=James L.|date=1992|pages=1–17|publisher=Springer-Verlag|language=en|doi=10.1007/bfb0036046|isbn=978-3540558514|title=Advanced Methods for Satellite and Deep Space Communications|volume=182|series=Lecture Notes in Control and Information Sciences|chapter=Deep-space communications and coding: A marriage made in heaven|citeseerx=10.1.1.36.4265}}[https://www.isiweb.ee.ethz.ch/archive/massey_pub/pdf/BI321.pdf pdf]</ref> Moreover, the proposed [[5G|5G standard]]<ref>{{cite web|url=http://www.3gpp.org/ftp/tsg_ran/WG1_RL1/TSGR1_87/Report/Final_Minutes_report_RAN1%2387_v100.zip|title=3GPP RAN1 meeting #87 final report|publisher=3GPP|accessdate=31 August 2017}}</ref> relies on the closely related [[Polar code (coding theory)|polar codes]]<ref>{{Cite journal|title=Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels - IEEE Journals & Magazine|journal=IEEE Transactions on Information Theory|volume=55|issue=7|pages=3051–3073|language=en-US|doi=10.1109/TIT.2009.2021379|year=2009|last1=Arikan|first1=Erdal|hdl=11693/11695|arxiv=0807.3917|s2cid=889822 }}</ref> 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 codes generalize the [[Reed–Solomon error correction|Reed–Solomon codes]] and the [[Hadamard code|Walsh–Hadamard code]]. Reed–Muller codes are [[Linear code|linear block codes]] that are [[locally testable code|locally testable]], [[locally decodable code|locally decodable]], and [[List decoding|list decodable]]. These properties make them particularly useful in the design of [[probabilistically checkable proof]]s.

Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When ''r'' and ''m'' are integers with 0 ≤ ''r'' ≤ ''m'', the Reed–Muller code with parameters  ''r'' and ''m'' is denoted as RM(''r'',&nbsp;''m''). When asked to encode a message consisting of ''k'' bits, where  <math>\textstyle k=\sum_{i=0}^r \binom{m}{i}</math> holds, the RM(''r'',&nbsp;''m'') code produces a codeword consisting of 2<sup>''m''</sup> bits.

Reed–Muller codes are named after [[David E. Muller]], who discovered the codes in 1954,<ref>{{Cite journal|last=Muller|first=David E.|date=1954|title=Application of Boolean algebra to switching circuit design and to error detection|journal=Transactions of the I.R.E. Professional Group on Electronic Computers|language=en-US|volume=EC-3|issue=3|pages=6–12|doi=10.1109/irepgelc.1954.6499441|issn=2168-1740}}</ref> and [[Irving S. Reed]], who proposed the first efficient decoding algorithm.<ref>{{Cite journal|last=Reed|first=Irving S.|date=1954|title=A class of multiple-error-correcting codes and the decoding scheme|journal=Transactions of the IRE Professional Group on Information Theory|language=en-US|volume=4|issue=4|pages=38–49|doi=10.1109/tit.1954.1057465|issn=2168-2690|hdl=10338.dmlcz/143797|hdl-access=free}}</ref>

==Description using low-degree polynomials==
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 code]]s and [[locally decodable code]]s.<ref>Prahladh Harsha et al., [[arxiv:1002.3864|Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)]], Section 5.2.1.</ref>

=== Encoder ===
A [[block code]] can have one or more encoding functions <math display="inline"> C:\{0,1\}^k\to\{0,1\}^{n} </math> that map messages <math display="inline"> x\in\{0,1\}^k </math> to codewords <math display="inline"> C(x)\in\{0,1\}^{n} </math>. The Reed&ndash;Muller code {{nowrap|RM(''r'', ''m'')}} has [[Block code#The message length k|message length]] <math>\textstyle k=\sum_{i=0}^r \binom{m}{i}</math> and [[Block code#The block length n|block length]]  <math>\textstyle n=2^m</math>. One way to define an encoding for this code is based on the evaluation of [[multilinear polynomial]]s with ''m'' variables and [[total degree]] at most ''r''. Every multilinear polynomial over the [[finite field]] with two elements can be written as follows:
<math display="block">p_c(Z_1,\dots,Z_m) = \sum_{\underset{|S|\le r}{S\subseteq\{1,\dots,m\}}} c_S\cdot \prod_{i\in S} Z_i\,.</math>
The <math display="inline"> Z_1,\dots,Z_m </math> are the variables of the polynomial, and the values <math display="inline"> c_S\in\{0,1\} </math> are the coefficients of the polynomial. Note that there are exactly <math display="inline"> k=\sum_{i=0}^r \binom{m}{i}</math> coefficients. With this in mind, an input message consists of <math display="inline"> k </math> values <math display="inline"> x\in\{0,1\}^k </math> which are used as these coefficients. In this way, each message <math display="inline"> x </math> gives rise to a unique polynomial <math display="inline"> p_x </math> in ''m'' variables. To construct the codeword <math display="inline"> C(x) </math>, the encoder evaluates the polynomial <math display="inline"> p_x </math> at all points <math display="inline"> Z=(Z_1,\ldots,Z_m)\in\{0,1\}^m </math>, where the polynomial is taken with multiplication and addition mod 2 <math display="inline">(p_x(Z)\bmod 2) \in \{0,1\}</math>. That is, the encoding function is defined via<math display="block">C(x) = \left(p_x(Z)\bmod 2\right)_{Z\in\{0,1\}^m}\,.</math>

The fact that the codeword <math>C(x)</math> suffices to uniquely reconstruct <math>x</math> follows from [[Lagrange polynomial|Lagrange interpolation]], which states that the coefficients of a polynomial are uniquely determined when sufficiently many evaluation points are given. Since <math>C(0)=0</math> and <math>C(x+y)=C(x)+C(y) \bmod 2</math> holds for all messages <math>x,y\in\{0,1\}^k</math>, the function <math>C</math> is a [[linear map]]. Thus the Reed&ndash;Muller code is a [[linear code]].

==== Example ====
For the code {{nowrap|RM(''2'', ''4'')}}, the parameters are as follows:

<math display="inline"> \begin{align}
r&=2\\
m&=4\\
k&=\textstyle\binom{4}{2}+\binom{4}{1}+\binom{4}{0}= 6+4+1=11\\
n&=2^m=16\\
\end{align} </math>

Let <math display="inline"> C:\{0,1\}^{11}\to\{0,1\}^{16} </math> be the encoding function just defined. To encode the string x = 1 1010 010101 of length 11, the encoder first constructs the polynomial <math display="inline"> p_x </math> in 4 variables:<math display="block">\begin{align}
p_x(Z_1,Z_2,Z_3,Z_4)
&= 1
+ (1\cdot Z_1 + 0\cdot Z_2 + 1\cdot Z_3 + 0\cdot Z_4)
+ (0\cdot Z_1 Z_2 + 1\cdot Z_1Z_3 + 0\cdot Z_1Z_4 + 1\cdot Z_2Z_3 + 0\cdot Z_2Z_4+ 1\cdot Z_3Z_4)\\
&=1+Z_1+Z_3+Z_1Z_3+Z_2Z_3+Z_3Z_4
\end{align}</math>Then it evaluates this polynomial at all 16 evaluation points (0101 means <math>Z_1=0, Z_2=1, Z_3=0, Z_4=1)</math>:
<math display="block">
p_x(0000)= 1,\;
p_x(0001)= 1,\;
p_x(0010)= 0,\;
p_x(0011)= 1,\;</math>

<math display="block">
p_x(0100)= 1,\;
p_x(0101)= 1,\;
p_x(0110)= 1,\;
p_x(0111)= 0,\;</math>

<math display="block">
p_x(1000)= 0,\;
p_x(1001)= 0,\;
p_x(1010)= 0,\;
p_x(1011)= 1,\;</math>

<math display="block">
p_x(1100)= 0,\;
p_x(1101)= 0,\;
p_x(1110)= 1,\;
p_x(1111)= 0\,.</math>As a result, C(1 1010 010101) = 1101 1110 0001 0010 holds.

=== Decoder ===
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.

The algorithm from Reed is based on the following property:
you start from the code word, that is a sequence of evaluation points from an unknown polynomial <math display="inline"> p_x </math> of <math display="inline"> {\mathbb F}_2[X_1,X_2,...,X_m] </math> of degree at most <math display="inline"> r </math> that you want to find. The sequence may contains any number of errors up to <math display="inline"> 2^{m-r-1}-1 </math> included.

If you consider a monomial <math display="inline"> \mu </math> of the highest degree <math display="inline"> d </math> in <math display="inline"> p_x </math> and sum all the evaluation points of the polynomial where all variables in <math display="inline"> \mu </math> have the values 0 or 1, and all the other variables have value 0, you get the value of the coefficient (0 or 1) of <math display="inline"> \mu </math> in <math display="inline"> p_x </math> (There are <math display="inline"> 2^d </math> such points). This is due to the fact that all lower monomial divisors of <math display="inline"> \mu </math> appears an even number of time in the sum, and only <math display="inline"> \mu </math> appears once.

To take into account the possibility of errors, you can also remark that you can fix the value of other variables to any value. So instead of doing the sum only once for other variables not in <math display="inline"> \mu </math> with 0 value, you do it <math display="inline"> 2^{m-d} </math> times for each fixed valuations of the other variables. If there is no error, all those sums should be equals to the value of the coefficient searched. 
The algorithm consists here to take the majority of the answers as the value searched. If the minority is larger than the maximum number of errors possible, the decoding step fails knowing there are too many errors in the input code.

Once a coefficient is computed, if it's 1, update the code to remove the monomial <math display="inline"> \mu </math> from the input code and continue to next monomial, in reverse order of their degree.

==== Example ====

Let's consider the previous example and start from the code. With <math display="inline"> r=2, m=4 </math> we can fix at most 1 error in the code.
Consider the input code as 1101 1110 0001 0110 (this is the previous code with one error).

We know the degree of the polynomial <math display="inline"> p_x </math> is at most <math display="inline"> r=2 </math>, we start by searching for monomial of degree 2.

* <math display="inline"> \mu=X_3X_4 </math>
** we start by looking for evaluation points with <math display="inline"> X_1=0, X_2=0, X_3\in\{0,1\}, X_4\in\{0,1\}</math>. In the code this is: <u>1101</u> 1110 0001 0110. The first sum is 1 (odd number of 1).
** we look for evaluation points with <math display="inline"> X_1=0, X_2=1, X_3\in\{0,1\}, X_4\in\{0,1\}</math>. In the code this is: 1101 <u>1110 </u>0001 0110. The second sum is 1.
** we look for evaluation points with <math display="inline"> X_1=1, X_2=0, X_3\in\{0,1\}, X_4\in\{0,1\}</math>. In the code this is: 1101 1110 <u>0001</u> 0110. The third sum is 1.
** we look for evaluation points with <math display="inline"> X_1=1, X_2=1, X_3\in\{0,1\}, X_4\in\{0,1\}</math>. In the code this is: 1101 1110 0001 <u>0110</u>. The third sum is 0 (even number of 1).
The four sums don't agree (so we know there is an error), but the minority report is not larger than the maximum number of error allowed (1), so we take the majority and the coefficient of <math display="inline"> \mu </math> is 1.

We remove <math display="inline"> \mu </math> from the code before continue : code : 1101 1110 0001 0110, valuation of <math display="inline"> \mu </math> is 0001000100010001, the new code is 1100 1111 0000 0111

* <math display="inline"> \mu=X_2X_4 </math>
** <u>11</u>00 <u>11</u>11 0000 0111. Sum is 0
** 11<u>00</u> 11<u>11</u> 0000 0111. Sum is 0
** 1100 1111 <u>00</u>00 <u>01</u>11. Sum is 1
** 1100 1111 00<u>00</u> 01<u>11</u>. Sum is 0
One error detected, coefficient is 0, no change to current code.
* <math display="inline"> \mu=X_1X_4 </math>
** <u>11</u>00 1111 <u>00</u>00 0111. Sum is 0
** 11<u>00</u> 1111 00<u>00</u> 0111. Sum is 0
** 1100 <u>11</u>11 0000 <u>01</u>11. Sum is 1
** 1100 11<u>11</u> 0000 01<u>11</u>. Sum is 0
One error detected, coefficient is 0, no change to current code.
* <math display="inline"> \mu=X_2X_3 </math>
** <u>1</u>1<u>0</u>0 <u>1</u>1<u>1</u>1 0000 0111. Sum is 1
** 1<u>1</u>0<u>0</u> 1<u>1</u>1<u>1</u> 0000 0111. Sum is 1
** 1100 1111 <u>0</u>0<u>0</u>0 <u>0</u>1<u>1</u>1. Sum is 1
** 1100 1111 0<u>0</u>0<u>0</u> 0<u>1</u>1<u>1</u>. Sum is 0
One error detected, coefficient is 1, valuation of <math display="inline"> \mu </math> is 0000 0011 0000 0011, current code is now 1100 1100 0000 0100.

* <math display="inline"> \mu=X_1X_3 </math>
** <u>1</u>1<u>0</u>0 1100 <u>0</u>0<u>0</u>0 0100. Sum is 1
** 1<u>1</u>0<u>0</u> 1100 0<u>0</u>0<u>0</u> 0100. Sum is 1
** 1100 <u>1</u>1<u>0</u>0 0000 <u>0</u>1<u>0</u>0. Sum is 1
** 1100 1<u>1</u>0<u>0</u> 0000 0<u>1</u>0<u>0</u>. Sum is 0
One error detected, coefficient is 1, valuation of <math display="inline"> \mu </math> is 0000 0000 0011 0011, current code is now 1100 1100 0011 0111.
* <math display="inline"> \mu=X_1X_2</math>
** <u>1</u>100 <u>1</u>100 <u>0</u>011 <u>0</u>111. Sum is 0
** 1<u>1</u>00 1<u>1</u>00 0<u>0</u>11 0<u>1</u>11. Sum is 1
** 11<u>0</u>0 11<u>0</u>0 00<u>1</u>1 01<u>1</u>1. Sum is 0
** 110<u>0</u> 110<u>0</u> 001<u>1</u> 011<u>1</u>. Sum is 0
One error detected, coefficient is 0, no change to current code.
We know now all coefficient of degree 2 for the polynomial, we can start mononials of degree 1. Notice that for each next degree, there are twice as much sums, and each sums is half smaller.
* <math display="inline"> \mu=X_4 </math>
** <u>11</u>00 1100 0011 0111. Sum is 0
** 11<u>00</u> 1100 0011 0111. Sum is 0
** 1100 <u>11</u>00 0011 0111. Sum is 0
** 1100  11<u>00</u> 0011 0111. Sum is 0
** 1100  1100 <u>00</u>11 0111. Sum is 0
** 1100  1100 00<u>11</u> 0111. Sum is 0
** 1100  1100 0011 <u>01</u>11. Sum is 1
** 1100  1100 0011 01<u>11</u>. Sum is 0
One error detected, coefficient is 0, no change to current code.
* <math display="inline"> \mu=X_3 </math>
** <u>1</u>1<u>0</u>0 1100 0011 0111. Sum is 1
** 1<u>1</u>0<u>0</u> 1100 0011 0111. Sum is 1
** 1100 <u>1</u>1<u>0</u>0 0011 0111. Sum is 1
** 1100 1<u>1</u>0<u>0</u> 0011 0111. Sum is 1
** 1100  1100 <u>0</u>0<u>1</u>1 0111. Sum is 1
** 1100  1100 0<u>0</u>1<u>1</u> 0111. Sum is 1
** 1100  1100 0011 <u>0</u>1<u>1</u>1. Sum is 1
** 1100  1100 0011 0<u>1</u>1<u>1</u>. Sum is 0
One error detected, coefficient is 1, valuation of <math display="inline"> \mu </math> is 0011 0011 0011 0011, current code is now 1111 1111 0000 0100.

Then we'll find 0 for <math display="inline"> \mu=X_2 </math>, 1 for <math display="inline"> \mu=X_1 </math> and the current code become 1111 1111 1111 1011. 

For the degree 0, we have 16 sums of only 1 bit. The minority is still of size 1, and we found <math display="inline"> p_x=1+X_1+X_3+X_1X_3+X_2X_3+X_3X_4</math> and the corresponding initial word 1 1010 010101

=== Generalization to larger alphabets via low-degree polynomials ===
Using low-degree polynomials over a finite field <math>\mathbb F</math> of size <math>q</math>, it is possible to extend the definition of Reed&ndash;Muller codes to alphabets of size <math>q</math>. Let <math>m</math> and <math>d</math> be positive integers, where <math>m</math> should be thought of as larger than <math>d</math>. To encode a message <math display="inline">x\in\mathbb F^k</math> of width <math>k=\textstyle\binom{m+d}{m}</math>, the message is again interpreted as an <math>m</math>-variate polynomial <math>p_x</math> of total degree at most <math>d</math> and with coefficient from <math>\mathbb F</math>. Such a polynomial indeed has <math>\textstyle\binom{m+d}{m}</math> coefficients. The Reed–Muller encoding of <math>x</math> is the list of all evaluations of <math>p_x(a)</math> over all <math>a\in\mathbb F^m</math>. Thus the block length is <math>n=q^m</math>.

== Description using a generator matrix ==
{{Confusing section|date=March 2011|reason=this section uses dense notation that is not explained well for most readers}}

A [[generator matrix]] for a Reed&ndash;Muller code {{nowrap|RM(''r'', ''m'')}} of length {{nowrap|1=''N'' = 2<sup>''m''</sup>}} can be constructed as follows. Let us write the set of all ''m''-dimensional binary vectors as:

:<math> X = \mathbb{F}_2^m = \{ x_1, \ldots, x_{N} \}.  </math>

We define in ''N''-dimensional space <math>\mathbb{F}_2^N</math> the [[indicator vector]]s

:<math>\mathbb{I}_A \in \mathbb{F}_2^N</math>

on subsets <math> A \subset X </math> by:

:<math>\left( \mathbb{I}_A \right)_i = \begin{cases} 1 & \mbox{ if } x_i \in A \\ 0 & \mbox{ otherwise} \\ \end{cases} </math>

together with, also in <math>\mathbb{F}_2^N</math>, the binary operation

:<math> w \wedge z = (w_1 \cdot z_1, \ldots , w_N \cdot z_N ), </math>

referred to as the ''wedge product'' (not to be confused with the [[wedge product]] defined in exterior algebra). Here, <math>w=(w_1,w_2,\ldots,w_N)</math> and <math>z=(z_1,z_2,\ldots, z_N)</math> are points in <math>\mathbb{F}_2^N</math> (''N''-dimensional binary vectors), and the operation <math>\cdot</math> is the usual multiplication in the field <math>\mathbb{F}_2</math>.

<math>\mathbb{F}_2^m</math> is an ''m''-dimensional vector space over the field <math>\mathbb{F}_2</math>, so it is possible to write

<math>(\mathbb{F}_2)^m = \{ (y_m, \ldots , y_1) \mid y_i \in \mathbb{F}_2 \} .</math>

We define in ''N''-dimensional space <math>\mathbb{F}_2^N</math> the following vectors with length <math> N: v_0 = (1,1,\ldots,1) </math> and

:<math> v_i = \mathbb{I}_{ H_i } ,</math>

where 1 ≤ i ≤ ''m'' and the ''H''<sub>''i''</sub> are [[hyperplane]]s in <math>(\mathbb{F}_2)^m</math> (with dimension {{nowrap|''m'' &minus; 1}}):

:<math>H_i = \{ y \in ( \mathbb{F}_2 ) ^m \mid y_i = 0 \} .</math>

=== The generator matrix ===
The Reed&ndash;Muller {{nowrap|RM(''r'', ''m'')}} code of order ''r'' and length ''N''&nbsp;=&nbsp;2<sup>''m''</sup> is the code generated by ''v''<sub>0</sub> and the wedge products of up to ''r'' of the ''v''<sub>''i''</sub>, {{nowrap|1 ≤ ''i'' ≤ ''m''}}  (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 {{nowrap|RM(''r'', ''m'')}} code, using vectors and their wedge product permutations up to ''r'' at a time <math>{v_0, v_1, \ldots, v_n, \ldots, (v_{i_1} \wedge v_{i_2}), \ldots (v_{i_1} \wedge v_{i_2} \ldots \wedge v_{i_r})}</math>, as the rows of the generator matrix, where {{nowrap|1 ≤ ''i''<sub>''k''</sub> ≤ ''m''}}.

=== Example 1 ===

Let ''m'' = 3. Then ''N'' = 8, and

:<math> X = \mathbb{F}_2^3 = \{ (0,0,0), (0,0,1), (0,1,0)  \ldots, (1,1,1) \}, </math>

and

:<math>
\begin{align}
v_0 & = (1,1,1,1,1,1,1,1) \\[2pt]
v_1 & = (1,0,1,0,1,0,1,0) \\[2pt]
v_2 & = (1,1,0,0,1,1,0,0) \\[2pt]
v_3 & = (1,1,1,1,0,0,0,0).
\end{align}
</math>

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

:<math> \{ v_0, v_1, v_2, v_3 \},\, </math>

or more explicitly by the rows of the matrix:

:<math>
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0
\end{pmatrix}
</math>

=== Example 2 ===

The RM(2,3) code is generated by the set:
:<math> \{ v_0, v_1, v_2, v_3, v_1 \wedge v_2, v_1 \wedge v_3, v_2 \wedge v_3 \} </math>

or more explicitly by the rows of the matrix:

:<math>
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}
</math>

===Properties===
The following properties hold:

# The set of all possible wedge products of up to ''m'' of the ''v''<sub>''i''</sub> form a basis for <math>\mathbb{F}_2^N</math>.
# The RM&thinsp;(''r'', ''m'') code has [[linear code|rank]]
#:<math> \sum_{s=0}^r {m \choose s}. </math>
# {{math|1=RM&thinsp;(''r'', ''m'') = RM&thinsp;(''r'', ''m ''&minus; 1) {{!}} RM&thinsp;(''r'' &minus; 1, ''m'' &minus; 1)}} where '|' denotes the [[bar product (coding theory)|bar product]] of two codes.
# {{math|1=RM&thinsp;(''r'', ''m'')}} has minimum [[Hamming weight]] 2<sup>''m'' &minus; ''r''</sup>.

====Proof====
{{Ordered list
|
There are

::<math> \sum_{s=0}^m { m \choose s } = 2^m = N </math>

such vectors and <math>\mathbb{F}_2^N</math> have dimension ''N'' so it is sufficient to check that the ''N'' vectors span; equivalently it is sufficient to check that <math>\mathrm{RM}(m, m) = \mathbb{F}_2^N</math>.

Let ''x'' be a binary vector of length ''m'', an element of ''X''.  Let (''x'')<sub>''i''</sub> denote the ''i''<sup>th</sup> element of ''x''.  Define

::<math>y_i = \begin{cases} v_i & \text{ if } (x)_i = 0 \\ v_0+v_i & \text{ if } (x)_i = 1 \\ \end{cases} </math>
where 1 ≤ ''i'' ≤''m''.

Then <math>\mathbb{I}_{ \{ x \} } = y_1 \wedge \cdots \wedge y_m </math>

Expansion via the distributivity of the wedge product gives <math> \mathbb{I}_{ \{ x \} } \in \mathrm{RM}(m,m) </math>. Then since the vectors <math>\{ \mathbb{I}_{ \{ x \} } \mid x \in X \} </math> span <math>\mathbb{F}_2^N</math> we have <math>\mathrm{RM}(m,n)=\mathbb{F}_2^N</math>.

|
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.

|
Omitted.

|4=
By induction.

:The {{math|1=RM(0,&nbsp;''m'')}} code is the repetition code of length {{math|1=''N''&nbsp;=2<sup>''m''</sup>}} and weight ''N'' = 2<sup>''m''&minus;0</sup> = 2<sup>''m''&minus;''r''</sup>. By '''1''' <math>\mathrm{RM}(m,n)=\mathbb{F}_2^n</math> and has weight 1 = 2<sup>0</sup> = 2<sup>''m''&minus;''r''</sup>.

The article [[bar product (coding theory)]] gives a proof that the weight of the bar product of two codes ''C''<sub>1</sub> , ''C''<sub>2</sub> is given by

::<math>\min \{ 2w(C_1), w(C_2) \} </math>

:If 0 < ''r'' < ''m'' and if {{Ordered list |list_style_type=lower-roman
|{{math|1=RM(''r'',''m''&nbsp;&minus;&nbsp;1)}} has weight {{math|1=2<sup>''m''&minus;1&minus;''r''</sup>}}
|{{math|1=RM(''r''&nbsp;&minus;&nbsp;1,''m''&nbsp;&minus;&nbsp;1)}} has weight {{math|1=2<sup>''m''&minus;1&minus;(''r''&minus;1)</sup> = 2<sup>''m''&minus;''r''</sup>}}
}}
:then the bar product has weight

::<math>\min \{ 2 \times 2^{m-1-r}, 2^{m-r} \} = 2^{m-r} .</math>
}}

=== Decoding RM codes ===
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 ''r''th 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''&nbsp;+&nbsp;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 construction==

A Reed–Muller code RM(''r,m'') exists for any integers <math>m \ge 0</math> and <math>0 \le r \le m</math>. RM(''m'', ''m'') is defined as the universe (<math>2^m,2^m,1</math>) code. RM(&minus;1,m) is defined as the trivial code (<math>2^m,0,\infty</math>). The remaining RM codes may be constructed from these elementary codes using the length-doubling construction

:<math>\mathrm{RM}(r,m) = \{(\mathbf{u},\mathbf{u}+\mathbf{v})\mid\mathbf{u} \in \mathrm{RM}(r,m-1),\mathbf{v} \in \mathrm{RM}(r-1,m-1)\}.</math>

From this construction, RM(''r,m'') is a binary [[linear block code]] (''n'', ''k'', ''d'') with length {{math|1=''n''&nbsp;=&nbsp;2<sup>''m''</sup>}}, dimension <math>k(r,m)=k(r,m-1)+k(r-1,m-1)</math> and minimum distance <math>d = 2^{m-r}</math> for <math>r \ge 0</math>. 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 {{math|1=''k''&nbsp;=&nbsp;''n''/2}} are self-dual.

==Special cases of Reed&ndash;Muller codes==

=== Table of all RM(r,m) codes for m≤5 ===
All {{math|1=RM(''r'',&nbsp;''m'')}} codes with <math>0\le m\le 5</math>  and alphabet size 2 are displayed here, annotated with the standard [n,k,d] [[Block code#Popular notation|coding theory notation]] for [[block code]]s. The code {{math|1=RM(''r'',&nbsp;''m'')}} is a <math>\textstyle [2^m,k,2^{m-r}]_2</math>-code, that is, it is a [[linear code]] over a [[binary set|binary alphabet]], has [[Block code#The block length n|block length]] <math>\textstyle 2^m</math>, [[Block code#The message length k|message length]] (or dimension) {{mvar|k}}, and [[Block code#The distance d|minimum distance]] <math>\textstyle 2^{m-r}</math>.

{|
|0
|1
|2
|3
|4
|5
|m
|
|-
|
|
|
|
|
|Z
|{{math|1=RM(''m,m'')}}<br />({{math|1=2<sup>''m''</sup>}}, {{math|1=2<sup>''m''</sup>}}, 1)
|universe codes
|-
|
|
|
|
|
|RM(5,5)<br />(32,32,1)
|
|
|-
|
|
|
|
|RM(4,4)<br />(16,16,1)
|
|{{math|1=RM(''m''&nbsp;&minus;&nbsp;1,&nbsp;''m'')}}<br />{{math|1=(2<sup>''m''</sup>, 2<sup>''m''</sup>&minus;1, 2)}}
|[[parity bit|SPC]] codes
|-
|
|
|
|RM(3,3)<br />(8,8,1)
|
|RM(4,5)<br />(32,31,2)
|-
|
|
|RM(2,2)<br />(4,4,1)
|
|RM(3,4)<br />(16,15,2)
|
|{{math|1=RM(''m''&nbsp;&minus;&nbsp;2, ''m'')}}<br />{{math|1=(2<sup>''m''</sup>, 2<sup>''m''</sup>&minus;''m''&minus;1, 4)}}
|extended [[Hamming code]]s
|-
|
|RM(1,1)<br />(2,2,1)
|
|RM(2,3)<br />(8,7,2)
|
|RM(3,5)<br />(32,26,4)
|-
|RM(0,0)<br />(1,1,1)
|
|RM(1,2)<br />(4,3,2)
|
|RM(2,4)<br />(16,11,4)
|-
|
|RM(0,1)<br />(2,1,2)
|
|RM(1,3)<br />(8,4,4)
|
|RM(2,5)<br />(32,16,8)
|{{math|1=RM(''r'', ''m''=2''r''+1)}}<br />{{math|1=(2<sup>2''r''+1</sup>, 2<sup>2''r''</sup>, 2<sup>''r''+1</sup>)}}
|[[self-dual code]]s
|-
|RM(&minus;1,0)<br />(1,0,<math>\infty</math>)
|
|RM(0,2)<br />(4,1,4)
|
|RM(1,4)<br />(16,5,8)
|-
|
|RM(−1,1)<br />(2,0,<math>\infty</math>)
|
|RM(0,3)<br />(8,1,8)
|
|RM(1,5)<br />(32,6,16)
|-
|
|
|RM(−1,2)<br />(4,0,<math>\infty</math>)
|
|RM(0,4)<br />(16,1,16)
|
|{{math|1=RM(1,''m'')}}<br />{{math|1=(2<sup>''m''</sup>, ''m''+1, 2<sup>''m''&minus;1</sup>)}}
|punctured [[Hadamard code]]s
|-
|
|
|
|RM(&minus;1,3)<br />(8,0,<math>\infty</math>)
|
|RM(0,5)<br />(32,1,32)
|-
|
|
|
|
|RM(&minus;1,4)<br />(16,0,<math>\infty</math>)
|
|{{math|1=RM(0,''m'')}}<br />{{math|1=(2<sup>''m''</sup>, 1, 2<sup>''m''</sup>)}}
|[[repetition code]]s
|-
|
|
|
|
|
|RM(&minus;1,5)<br />(32,0,<math>\infty</math>)
|-
|
|
|
|
|
|
|{{math|1=RM(&minus;1,''m'')}}<br />{{math|1=(2<sup>''m''</sup>, 0, &infin;)}}
|trivial codes
|}

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

*{{math|1=RM(0,&nbsp;''m'')}} codes are [[repetition code]]s of length {{math|1=''N''&nbsp;=&nbsp;2<sup>''m''</sup>}}, [[code rate|rate]] <math>{R=\tfrac{1}{N}}</math> and minimum distance <math>d_\min = N</math>.
*{{math|1=RM(1,&nbsp;''m'')}} codes are [[Low-density parity-check code|parity check codes]] of length {{math|1=''N''&nbsp;=&nbsp;2<sup>''m''</sup>}}, rate <math>R=\tfrac{m+1}{N}</math> and minimum distance <math>d_\min = \tfrac{N}{2}</math>.
*{{math|1=RM(''m''&nbsp;&minus;&nbsp;1,&nbsp;''m'')}} codes are [[Low-density parity-check code|single parity check codes]] of length {{math|1=''N''&nbsp;=&nbsp;2<sup>''m''</sup>}}, rate <math>R=\tfrac{N-1}{N}</math> and minimum distance <math>d_\min = 2</math>.
*{{math|1=RM(''m''&nbsp;&minus;&nbsp;2,&nbsp;''m'')}} codes are the family of [[Hamming code|extended Hamming codes]] of length {{math|1=''N''&nbsp;=&nbsp;2<sup>''m''</sup>}} with minimum distance <math>d_\min = 4</math>.<ref>Trellis and Turbo Coding, C. Schlegel & L. Perez, Wiley Interscience, 2004, p149.</ref>

== References ==
{{reflist}}

== Further reading ==

* {{cite book | author=Shu Lin |author2=Daniel Costello  | title=Error Control Coding | edition=2 | year=2005 | publisher=Pearson | isbn=978-0-13-017973-9 }} Chapter 4.
* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2 | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | year=1992 | isbn=978-3-540-54894-2 | url-access=registration | url=https://archive.org/details/introductiontoco0000lint }} Chapter 4.5.

==External links==
* [http://ocw.mit.edu MIT OpenCourseWare], 6.451 Principles of Digital Communication II, Lecture Notes section 6.4
* [http://octave.sourceforge.net/communications/function/reedmullergen.html GPL Matlab-implementation of RM-codes ]
* [http://octave.svn.sourceforge.net/viewvc/octave/trunk/octave-forge/main/comm/inst/reedmullergen.m?revision=9852&view=markup Source GPL Matlab-implementation of RM-codes]
*{{cite journal |last1=Weiss |first1=E. |title=Generalized Reed-Muller codes |journal=Information and Control |date=September 1962 |volume=5 |issue=3 |pages=213–222 |doi=10.1016/s0019-9958(62)90555-7 |issn=0019-9958|doi-access= }}

{{CCSDS}}

{{DEFAULTSORT:Reed-Muller code}}
[[Category:Error detection and correction]]
[[Category:Coding theory]]
[[Category:Theoretical computer science]]