Critical exponent

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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 Hadamard code, the Walsh–Hadamard code, and the Reed–Solomon code.

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

RM(0, r) codes are repetition codes of length n = 2 ^r, rate $R=\frac{1}{n}$ and minimum distance d_min = n.

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

RM(r − 1, r) codes are parity check codes of length n = 2 ^r.

RM(r − 2, r) codes are the family of extended Hamming codes of length n = 2 ^r with minimum distance d_min = 4.

Construction

A generator matrix for a Reed–Muller code of length n = 2^d can be constructed as follows. Let us write:

$X={\mathbb{𝔽}}_2^d=\{x_1,\dots ,x_{2^d}\}.$

Note that each member of the set X is a point in ${\mathbb{𝔽}}_2^d$. 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:

${\left({\mathbb{𝕀}}_A\right)}_i=\{\begin{array}{ll}1& \text{ if }{x}_i\in A\\ 0& \text{ otherwise}\\ \end{array}$

together with, also in ${\mathbb{𝔽}}_2^n$, the binary operation

$w\wedge z=(w_1\cdot z_1,\dots ,w_n\cdot z_n),$

referred to as the wedge product (this wedge product is not to be confused with the wedge product defined in exterior algebra). Here, $w=(w_1,w_2,\dots ,w_n)$ and $z=(z_1,z_2,\dots ,z_n)$ are points in ${\mathbb{𝔽}}_2^n$, and the operation $\cdot$ is the usual multiplication in the field ${\mathbb{𝔽}}_2$.

${\mathbb{𝔽}}_2^d$ is a d-dimensional vector space over the field ${\mathbb{𝔽}}_2$, so it is possible to write

$({\mathbb{𝔽}}_2{)}^d=\{(y_d,\dots ,y_1)|y_i\in {\mathbb{𝔽}}_2\}$

We define in n-dimensional space ${\mathbb{𝔽}}_2^n$ the following vectors with length n: v₀ = (1, 1, 1, 1, 1, 1, 1, 1) and

$v_i={\mathbb{𝕀}}_{H_i}$

where the H_i are hyperplanes in $({\mathbb{𝔽}}_2{)}^d$ (with dimension d −1):

$H_i=\{y\in ({\mathbb{𝔽}}_2{)}^d\mid y_i=0\}$

Building a generator matrix

The Reed–Muller RM(d, r) code of order d and length n = 2^r is the code generated by v₀ and the wedge products of up to r of the v_i (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 RM(d,r) code, using vectors and their wedge product permutations up to the r at a time $v_0,v_1,\dots ,v_d,\dots ,(v_i\wedge v_j),\dots (v_i\wedge v_j\dots \wedge v_r)$, as the rows of the generator matrix.

Example 1

Let r = 3. Then n = 8, and

$X={\mathbb{𝔽}}_2^3=\{(0,0,0),(0,0,1),\dots ,(1,1,1)\},$

and

$\begin{array}{ccc}{v}_0& =& (1,1,1,1,1,1,1,1)\\ v_1& =& (1,0,1,0,1,0,1,0)\\ v_2& =& (1,1,0,0,1,1,0,0)\\ v_3& =& (1,1,1,1,0,0,0,0).\\ \end{array}$

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

$\left(\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& 0& 1& 0& 1& 0& 1& 0\\ 1& 1& 0& 0& 1& 1& 0& 0\\ 1& 1& 1& 1& 0& 0& 0& 0\\ \end{array}\right)$

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:

$\left(\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& 0& 1& 0& 1& 0& 1& 0\\ 1& 1& 0& 0& 1& 1& 0& 0\\ 1& 1& 1& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0\\ \end{array}\right)$

Properties

The following properties hold:

1 The set of all possible wedge products of up to d of the v_i form a basis for ${\mathbb{𝔽}}_2^n$.

2 The RM (d, r) code has rank

$\sum _{s=0}^r(\genfrac{}{}{0ex}{}{d}{s}).$

3 RM (d, r) = RM (d− 1, r ) | RM (d − 1, r − 1) where '|' denotes the bar product of two codes.

4 RM (d, r) has minimum Hamming weight 2^{d − r}.

Proof

1

There are

$\sum _{s=0}^d(\genfrac{}{}{0ex}{}{d}{s})=2^d=n$

such vectors and ${\mathbb{𝔽}}_2^n$ has dimension n so it is sufficient to check that the n vectors span; equivalently it is sufficient to check that RM(d, d) = ${\mathbb{𝔽}}_2^n$.

Let x be an element of X and define

$y_i=\{\begin{array}{ll}{v}_i& \text{ if }{x}_i=0\\ 1+v_i& \text{ if }{x}_i=1\\ \end{array}$

Then ${\mathbb{𝕀}}_{\{x\}}=y_i\wedge \dots \wedge y_d$

Expansion via the distributivity of the wedge product gives ${\mathbb{𝕀}}_{\{x\}}\in \text{ RM(d,d)}$. Then since the vectors $\{{\mathbb{𝕀}}_{\{x\}}\mid x\in X\}$ span ${\mathbb{𝔽}}_2^n$ we have RM(d, d) = ${\mathbb{𝔽}}_2^n$.

2

By 1, all such wedge products must be linearly independent, so the rank of RM(d, r) must simply be the number of such vectors.

3

Omitted.

4

By induction.

The RM(d, 0) code is the repetition code of length n =2^d and weight n = 2^d−0 = 2^d−0. By 1 RM(d, d) = ${\mathbb{𝔽}}_2^n$ and has weight 1 = 2⁰ = 2^d−d.

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 < d and if

i) RM(d − 1, r) has weight 2^d−1−r

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

then the bar product has weight

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

Alternative construction

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

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

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 $r\ge 0$. 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.

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 $(\genfrac{}{}{0ex}{}{m+d}{m})$ 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 $(\genfrac{}{}{0ex}{}{m+d}{m})$ 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.

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

Decoding RM codes

RM(r, m) codes can be decoded using the 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.

Notes

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References

Research Articles:

• D. E. Muller. Application of boolean algebra to switching circuit design and to error detection. IRE Transactions on Electronic Computers, 3:6–12, 1954.
• Irving S. Reed. A class of multiple-error-correcting codes and the decoding scheme. Transactions of the IRE Professional Group on Information Theory, 4:38–49, 1954.

Textbooks:

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 4.
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 4.5.

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