Reed–Muller code

Reed-Muller code RM(r,m)
Named after	Irving S. Reed and David E. Muller
Classification
Type	Linear block code
Block length	$2^{m}$
Message length	$k=\sum _{{i=0}}^{r}{\binom {m}{i}}$
Rate	$k/2^{m}$
Distance	$2^{{m-r}}$
Alphabet size	$2$
Notation	$[2^{m},k,2^{{m-r}}]_{2}$ -code

Reed–Muller codes are a family of linear error-correcting codes used in communications. Reed–Muller codes belong to the classes of locally testable codes and locally decodable codes, which is why they are useful in the design of probabilistically checkable proofs in computational complexity theory. They are named after Irving S. Reed and David E. Muller. Muller discovered the codes, and Reed proposed the majority logic decoding for the first time. Special cases of Reed–Muller codes include the Walsh–Hadamard code and the Reed–Solomon code.

Reed–Muller codes are listed as RM(r, m), where r is the order of the code, 0 ≤ r ≤ m, and m determines the block length N = 2^m. RM codes are related to binary functions on the field GF(2^m) over the elements {0, 1}.

RM(0, m) codes are repetition codes of length N = 2^m, rate ${R={\tfrac {1}{N}}}$ and minimum distance $d_{\min }=N$ .

RM(1, m) codes are parity check codes of length N = 2^m, rate $R={\tfrac {m+1}{N}}$ and minimum distance $d_{\min }={\tfrac {N}{2}}$ .

RM(m − 1, m) codes are single parity check codes of length N = 2^m, rate $R={\tfrac {N-1}{N}}$ and minimum distance $d_{\min }=2$ .

RM(m − 2, m) codes are the family of extended Hamming codes of length N = 2^m with minimum distance $d_{\min }=4$ .^[1]

Construction

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

X=\mathbb {F} _{2}^{m}=\{x_{1},\ldots ,x_{N}\}.

We define in N-dimensional space $\mathbb {F} _{2}^{N}$ the indicator vectors

\mathbb {I} _{A}\in \mathbb {F} _{2}^{N}

on subsets $A\subset X$ by:

\left({\mathbb {I}}_{A}\right)_{i}={\begin{cases}1&{\mbox{ if }}x_{i}\in A\\0&{\mbox{ otherwise}}\\\end{cases}}

together with, also in $\mathbb {F} _{2}^{N}$ , the binary operation

w\wedge z=(w_{1}\cdot z_{1},\ldots ,w_{N}\cdot z_{N}),

referred to as the wedge product (this wedge product is not to be confused with the wedge product defined in exterior algebra). Here, $w=(w_{1},w_{2},\ldots ,w_{N})$ and $z=(z_{1},z_{2},\ldots ,z_{N})$ are points in $\mathbb {F} _{2}^{N}$ (N-dimensional binary vectors), and the operation $\cdot$ is the usual multiplication in the field $\mathbb {F} _{2}$ .

$\mathbb {F} _{2}^{m}$ is an m-dimensional vector space over the field $\mathbb {F} _{2}$ , so it is possible to write

$(\mathbb {F} _{2})^{m}=\{(y_{m},\ldots ,y_{1})\mid y_{i}\in \mathbb {F} _{2}\}$

We define in N-dimensional space $\mathbb {F} _{2}^{N}$ the following vectors with length $N:v_{0}=(1,1,\ldots ,1)$ and

v_{i}={\mathbb {I}}_{{H_{i}}}

where 1 ≤ i ≤ m and the H_i are hyperplanes in $(\mathbb {F} _{2})^{m}$ (with dimension m −1):

H_{i}=\{y\in (\mathbb {F} _{2})^{m}\mid y_{i}=0\}

Building a generator matrix

The Reed–Muller RM(r, m) code of order r and length N = 2^m is the code generated by v₀ and the wedge products of up to r of the v_i, 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 RM(r,m) code, using vectors and their wedge product permutations up to r at a time ${v_{0},v_{1},\ldots ,v_{n},\ldots ,(v_{i_{1}}\wedge v_{i_{2}}),\ldots (v_{i_{1}}\wedge v_{i_{2}}\ldots \wedge v_{i_{r}})}$ , as the rows of the generator matrix, where 1 ≤ i_k ≤ m.

Example 1

Let m = 3. Then N = 8, and

X={\mathbb {F}}_{2}^{3}=\{(0,0,0),(0,0,1),\ldots ,(1,1,1)\},

and

{\begin{aligned}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{aligned}}

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

\{v_{0},v_{1},v_{2},v_{3}\},\,

or more explicitly by the rows of the matrix:

{\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}}

Example 2

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

\{v_{0},v_{1},v_{2},v_{3},v_{1}\wedge v_{2},v_{1}\wedge v_{3},v_{2}\wedge v_{3}\}

or more explicitly by the rows of the matrix:

{\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}}

Properties

The following properties hold:

1 The set of all possible wedge products of up to m of the v_i form a basis for $\mathbb {F} _{2}^{N}$ .

2 The RM (r, m) code has rank

\sum _{s=0}^{r}{m \choose s}.

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 2^{m − r}.

Proof

There are

\sum _{s=0}^{m}{m \choose s}=2^{m}=N

such vectors and

\mathbb {F} _{2}^{N}

have dimension N so it is sufficient to check that the N vectors span; equivalently it is sufficient to check that RM(m, m) =

\mathbb {F} _{2}^{N}

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

y_{i}={\begin{cases}v_{i}&{\text{ if }}(x)_{i}=0\\v_{0}+v_{i}&{\text{ if }}(x)_{i}=1\\\end{cases}}

where 1 ≤ i ≤m.

Then

\mathbb {I} _{\{x\}}=y_{1}\wedge \cdots \wedge y_{m}

Expansion via the distributivity of the wedge product gives

\mathbb {I} _{\{x\}}\in {\mbox{ RM(m,m)}}

. Then since the vectors

\{{\mathbb {I}}_{{\{x\}}}\mid x\in X\}

span

\mathbb {F} _{2}^{N}

we have RM(m, m) =

\mathbb {F} _{2}^{N}

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.

By induction.

The RM(0, m) code is the repetition code of length N =2^m and weight N = 2^m−0 = 2^m−r. By 1 RM(m, m) =

{\mathbb {F}}_{2}^{n}

and has weight 1 = 2⁰ = 2^m−r.

The article bar product (coding theory) gives a proof that the weight of the bar product of two codes C₁ , C₂ is given by

\min\{2w(C_{1}),w(C_{2})\}

If 0 < r < m and if

i) RM(r ,m − 1) has weight 2^m−1−r

ii) RM(r-1,m-1) has weight 2^{m−1−(r−1)} = 2^m−r

then the bar product has weight

\min\{2\times 2^{m-1-r},2^{m-r}\}=2^{m-r}.

Alternative construction

A Reed–Muller code RM(r,m) exists for any integers $m\geq 0$ and $0\leq r\leq m$ . RM(m, m) is defined as the universe ( $2^{m},2^{m},1$ ) code. RM(−1,m) is defined as the trivial code ( $2^{m},0,\infty$ ). The remaining RM codes may be constructed from these elementary codes using the length-doubling construction

RM(r,m)=\{(\mathbf {u} ,\mathbf {u} +\mathbf {v} )\mid \mathbf {u} \in RM(r,m-1),\mathbf {v} \in RM(r-1,m-1)\}.

From this construction, RM(r,m) is a binary linear block code (n, k, d) with length n = 2^m, dimension $k(r,m)=k(r,m-1)+k(r-1,m-1)$ and minimum distance $d=2^{{m-r}}$ for $r\geq 0$ . 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.

Construction based on low-degree polynomials over a finite field

There is another, simple way to construct Reed–Muller codes based on low-degree polynomials over a finite field. This construction is particularly suited for their application as locally testable codes and locally decodable codes.^[2]

Let $\mathbb F$ be a finite field and let $m$ and $d$ be positive integers, where $m$ should be thought of as larger than $d$ . We are going to encode messages consisting of ${m+d \choose m}$ elements of $\mathbb F$ as codewords of length $|{\mathbb F}|^{m}$ as follows: We interpret the message as an $m$ -variate polynomial $f$ of degree at most $d$ with coefficient from $\mathbb F$ . Such a polynomial has ${m+d \choose m}$ coefficients. The Reed–Muller encoding of $f$ is the list of the evaluations of $f$ on all $x\in {\mathbb F}^{m}$ ; the codeword at the position indexed by $x\in {\mathbb F}^{m}$ has value $f(x)$ .

Table of Reed–Muller codes

The table below lists the RM(r, m) codes of lengths up to 32 for alphabet of size 2 annotated with standard coding theory notation for block codes The Reed–Muller code is a $[2^{m},k,2^{{m-r}}]_{2}$ -code, that is, it is a linear code over a binary alphabet, has block length $2^{m}$ , message length (or dimension) $k$ , and minimum distance $2^{{m-r}}$ .

						RM(m,m) ( $2^{m},2^{m},1$ )	universe codes
					RM(5,5) (32,32,1)
				RM(4,4) (16,16,1)		RM(m − 1, m) ( $2^{m},2^{m}-1,2$ )	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) ( $2^{m},2^{m}-m-1,4$ )	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, $\infty$ )		RM(0,2) (4,1,4)		RM(1,4) (16,5,8)
	RM(-1,1) (2,0, $\infty$ )		RM(0,3) (8,1,8)		RM(1,5) (32,6,16)
		RM(-1,2) (4,0, $\infty$ )		RM(0,4) (16,1,16)		RM(1,m) ( $2^{m},m+1,2^{{m-1}}$ )	Punctured hadamard codes
			RM(−1,3) (8,0, $\infty$ )		RM(0,5) (32,1,32)
				RM(−1,4) (16,0, $\infty$ )		RM(0,m) ( $2^{m},1,2^{m}$ )	repetition codes
					RM(−1,5) (32,0, $\infty$ )
						RM(-1,m) ( $2^{m},0,\infty$ )	trivial codes

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 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.

References

↑ Trellis and Turbo Coding, C. Schlegel & L. Perez, Wiley Interscience, 2004, p149.
↑ Prahladh Harsha et al., Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes), Section 5.2.1.

External links

MIT OpenCourseWare, 6.451 Principles of Digital Communication II, Lecture Notes section 6.4
GPL Matlab-implementation of RM-codes
Source GPL Matlab-implementation of RM-codes

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