Reed–Muller code

Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes 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. For example, they have been shown to asymptotically achieve Shannon capacity on symmetric memoryless channels.
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 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM. When asked to encode a message consisting of k bits, where holds, the RM code produces a codeword consisting of 2^m bits.
Reed–Muller codes are named after David E. Muller, who discovered the codes in 1954, and Irving S. Reed, who proposed the first efficient decoding algorithm.

Description using low-degree polynomials

Reed–Muller codes can be described in several different 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.

Encoder

A block code can have one or more encoding functions that map messages to codewords. The Reed-Muller code 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 at most 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. Note that there are exactly coefficients. With this in mind, an input message consists of values which are 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 the polynomial at all points, where the polynomial is taken with multiplication and addition mod 2. 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.

Example

For the code, 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 = 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 of of degree at most that you want to find. The sequence may contains any number of errors up to included.
If you consider a monomial of the highest degree in and sum all the evaluation points of the polynomial where all variables in have the values 0 or 1, and all the other variables have value 0, you get the value of the coefficient of in . This is due to the fact that all lower monomial divisors of appears an even number of time in the sum, and only 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 with 0 value, you do it 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 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 we can fix at most 1 error in the code.
Consider the input code as 1101 1110 0001 0110.
We know the degree of the polynomial is at most, we start by searching for monomial of degree 2.

* we start by looking for evaluation points with. In the code this is: 1101 1110 0001 0110. The first sum is 1.
* we look for evaluation points with. In the code this is: 1101 1110 0001 0110. The second sum is 1.
* we look for evaluation points with. In the code this is: 1101 1110 0001 0110. The third sum is 1.
* we look for evaluation points with. In the code this is: 1101 1110 0001 0110. The third sum is 0.

The four sums don't agree, but the minority report is not larger than the maximum number of error allowed, so we take the majority and the coefficient of is 1.
We remove from the code before continue : code : 1101 1110 0001 0110, valuation of is 0001000100010001, the new code is 1100 1111 0000 0111

* 1100 1111 0000 0111. Sum is 0
* 1100 1111 0000 0111. Sum is 0
* 1100 1111 0000 0111. Sum is 1
* 1100 1111 0000 0111. Sum is 0

One error detected, coefficient is 0, no change to current code.

* 1100 1111 0000 0111. Sum is 0
* 1100 1111 0000 0111. Sum is 0
* 1100 1111 0000 0111. Sum is 1
* 1100 1111 0000 0111. Sum is 0

One error detected, coefficient is 0, no change to current code.

* 1100 1111 0000 0111. Sum is 1
* 1100 1111 0000 0111. Sum is 1
* 1100 1111 0000 0111. Sum is 1
* 1100 1111 0000 0111. Sum is 0

One error detected, coefficient is 1, valuation of is 0000 0011 0000 0011, current code is now 1100 1100 0000 0100.

* 1100 1100 0000 0100. Sum is 1
* 1100 1100 0000 0100. Sum is 1
* 1100 1100 0000 0100. Sum is 1
* 1100 1100 0000 0100. Sum is 0

One error detected, coefficient is 1, valuation of is 0000 0000 0011 0011, current code is now 1100 1100 0011 0111.

* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. 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.

* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 0
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 0

One error detected, coefficient is 0, no change to current code.

* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 1
* 1100 1100 0011 0111. Sum is 0

One error detected, coefficient is 1, valuation of is 0011 0011 0011 0011, current code is now 1111 1111 0000 0100.
Then we'll find 0 for, 1 for 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 and the corresponding initial word 1 1010 010101

Generalization to larger alphabets via low-degree polynomials

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 width, the message is again interpreted as 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 matrix

A generator matrix for a Reed-Muller code of length 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. Here, and are points in , 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 H_i are hyperplanes in :