Download ICTCM2006 - Radford University

Elementary Coding Theory
Including Hamming and
Reed-Solomom Codes with
Maple and MATLAB
Richard Klima
Neil Sigmon
Appalachian State University
Boone, North Carolina
Radford University
Radford, Virginia
Purpose for Presentation

To demonstrate some applications with
error correcting codes with MATLAB
and Maple



Reed-Muller Codes
Hamming Codes
Reed-Solomon Codes
Basic Notation Terminology

Coding theory is the study of methods for
efficient and accurate transformation from
one place to another.

A code is a set of codewords representing
the messages to be transferred between
two parties.

An error-correcting code is a code where
it is possible to detect and correct errors
during the transmission of codewords.

Applications




Information transmitted by Internet.
Wireless communication devices.
Compact Discs.
Transmission of information from satellites.
Reed-Muller Codes

Invented by I. Reed and D. Muller in 1954.

Can be constructed using matrices.

Can correct a multiple number of errors.

Was used to perform error correction for
the Mariner 9 space probe mission to Mars
in 1972.
Hadamard Matrices

A n n matrix H is a Hadamard matrix if
all entries of H are 1 or -1 and HH T  nI ,
where I is the

n n
identity matrix.
A normalized Hadamard matrix is one
where the first row and column of H only
contain positive ones.

The only normalized Hadamard matrices
of order one and two are
H1  1

1 1 
and H 2  

1

1


Subsequent normalized Hadamard
matrices can be generated using the block
matrix formula
H n
H 2n  
H n
Hn 
 H n 
Reed-Muller Code Construction Steps

Construct a normalized Hadamard matrix
of order 4k  8 .

Delete the first row and first column of the
normalized Hadamard matrix. Change all
negative ones to zeros. Call this matrix A.

Change all ones to zeros and zeros to
ones in the matrix A. Call this matrix B.

Attach a column of all ones to the left of
the matrix A and a column of all zeros to
the left of the matrix B. Call these matrices
scriptA and scriptB, respectively.

Stack the matrix scriptA on top of scriptB.
Add an extra row of all zeros and an extra
row of all ones to the bottom of the matrix.

The codewords for the code are vectors
obtained from the rows for the resulting
matrix.

Each codeword c is a vector consisting
over elements in Z 2 of length 4k.

Reed-Muller codes are classified as
(4k, 2k) codes, where 4k is the length of
each codeword and 2k is the minimum
distance between codewords.

A Reed-Muller code has 8k codewords.
Error Correction for Reed-Muller Codes

Reed-Muller codes are designed to correct
up to k – 1 errors.

Suppose a vector r is received. This
vector is compared to all codewords in the
code. The vector r will be corrected to the
codeword that differs from it in k – 1 or
less positions.
MATLAB Example

We generate the Reed-Muller code that
was used for in Mariner 9 space probe
used to take pictures of Mars in 1972.

The code was constructed using the
normalized Hadamard H 32 (k = 8). The
code consisted of 64 codewords of length
32 each. The code could correct up to 7
errors.

Before being transmitted, each photograph
taken was broken down into a collection of
pixels.

Each pixel was assigned one of 64 levels
of grayness, and then encoded into one of
the 64 codewords in the code.
Hamming Codes

First published by Richard Hamming in
1950.

Used by the telephone industry and in
computer data transfer.

Can correct 1 error.

Code is perfect, which says every message
received with errors is uniquely correctable
to a codeword.
Hamming Codeword Construction

Let W
and V  Z 2n be vector
spaces with k  n and let G be a k  n
matrix over Z 2 of full row rank. Then
k
 Z2
C  {v V | v  wG for some w W }
is a subspace of
V of dimension k .
The vectors in C form 2 k codewords.

The matrix G is called the generator matrix
for the code C.

The matrix G is formed by finding the
nullspace of a matrix H known as the parity
check matrix.

Hamming codes form a linear [n, k] code,
where the codewords have a length of
n positions and k is the dimension of the
vector space the code forms.

In Hamming codes, n  2  1 and
m
k  2  1  m for some integer m > 1.
m

The parity check matrix H is a matrix of
m
size m  (2  1) that is formed by placing as
its columns the binary expressions of the
m
integers 1,2, ,2  1 .

The generator matrix G will be of size
(2m  1  m)  (2m  1).

If c  wG a codeword, then Hc  0 .
T
MATLAB Example

In MATLAB, suppose we demonstrate how to
construct a codeword from the vector
w = (10111011110).

We encode this message using a [15, 11]
Hamming code with generator matrix G
formed from the nullspace of the rows of the
parity check matrix
0
0
H 
0
1

0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 1 1 1 1 0 0 0 0 1 1 1 1

1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1
Error Correction for Hamming Codes

Suppose a vector r is transmitted and
T
received. If Hr  0, then r is a codeword.

If not, then the error can be corrected by
computing c  r  ei where ei is a vector
containing all zeros except at the i th
position.

The error position i is determined by using
T
T
the fact that Hr  Hei .
T
 Since Hei and
T produce the same
Hr
column, the column number produced by
T
Hr gives the error position i.
MATLAB Example

Using a [15, 11] Hamming code, we
demonstrate how to correct the following
received vector.
r = (011100010111110)
Reed-Solomon Codes

Developed by Irving Reed and Gustave
Solomon in 1960.

One of the most widely used error
correcting code techniques.

Ideal at correcting error bursts, where
several errors occur close together.

Important for the Voyager 2 satellite’s
transmission of pictures of the outer
planets Uranus and Neptune.

Besides satellite communications, Reed
Solomon codes are used with wireless
communication and with compact disks.
Example Pictures taken by Voyager
Pictures of Uranus and its rings
Example Pictures taken by Voyager
Picture of Neptune and one its moons, Triton
Codeword Construction in
Reed-Solomon Codes

The Reed-Solomon code implemented on
Voyager 2 was constructed using a finite
field F  Z 2 [ x] /( p( x)) of 28  256 elements
represented by the primitive polynomial
p ( x)  x8  x 4  x 3  x 2  1

Codewords for Voyager were obtained by
taking multiples of the generating polynomial
g ( x)  ( x  a)( x  a )( x  a )
2
32
where a is a primitive element use to generate F .

A codeword c (x ) is formed by computing the
polynomial
c ( x )  m( x ) g ( x )
where m(x) is a polynomial with coefficients
from the finite field F .

Each codeword c (x ) is represented by
polynomials of degree less than 28  1  255 .

Each codeword is a vector of length 255 with
the information transformed in the
polynomial’s coefficients.

Each polynomial coefficient in the codeword
is a polynomial of degree < p( x)  8. Each
polynomial coefficient is a unique element
from the finite field F .

On Voyager, full color images were digitized
into binary vectors over Z 2 .

The binary vectors can be represented in
“blocks” of length 8 as the coefficients of
polynomials of degree < p(x) = 8. For
example,
(10111001)  1  a  a  a  a
2

3
4
7
In binary, the length of each codeword
converts to a vector of length of
(deg p( x))  (28  1)  2040
Error Correction in
Reed-Solomon Codes

A codeword c (x ) has the property that
i
c(a )  0 for i  1,,32 . This gives a
technique for verifying a transmitted
message is a codeword.

If a transmitted message r(x) is received
that is not a codeword, a polynomial e(x) is
computed to correct its coefficients by
forming r ( x)  e( x)  c( x) .

The Voyager 2 Reed-Solomon code could
correct up to 32 / 2  16 polynomial
coefficient errors. This translates to a
minimum of 121 errors in binary.

Finding e(x) involves performing
computations involving the Euclidean
algorithm and polynomial differentiation.
Maple Example

Involves user-written procedures

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

bincoeff
binword
rscoeff
rseuclid
We demonstrate both the codeword
construction and error correction process.
References

R. E. Klima, N. P. Sigmon, and E. L. Stitzinger,
Applications of Abstract Algebra with Maple, CRC
Press, Llc, Boca Raton, FL 2000.

Stephen B. Wicker and Vijay K Bhargava, editors:
Reed-Solomon Codes and their Applications, The
Institute of Electrical and Electronics Engineers,
Inc., New York, NY 1994.