Page 336 - Kaleidoscope Academic Conference Proceedings 2024

2024 ITU Kaleidoscope Academic Conference

each iteration, and the mean of the messages from CN and The big bang in the field of algebraic code was the
BN after l iterations is termed as ( ) and , ( ) , invention of RS codes in 1960 [7].
,
respectively
The RS code is a type of non-binary cyclic ECCs defined
The PDF of in iteration is thus modeled considering over the finite field and can correct multiple random errors.
ℎ
the CNs aggregate the messages from several BNs, the This class of codes was first discovered by Irving S. Reed
messages from the CNs can thus be assumed to be Gaussian and Gustave Solomon in 1960. In RS codes, we begin by
specifying the number of random errors desired to be
corrected by the code. Then we construct the generator
( ( ), 2 ( ), )
,
,
polynomial, ( ) for the RS codes.
= ( ( , 2 )

∗ (( − 1) ( Firstly, we will develop knowledge of some necessary

(6
− 1), 2( − 1) ( mathematical tools required for the study of RS codes.

) Then later, we will look into the encoding procedure for RS
− 1))) ( ),
codes following which the efficient RS decoding technique
will follow. Here, we list some interesting facts about the
finite field for the construction of RS codes:
a) For any , the number of elements must be the

power of a prime. For example and exist but
2 5 8

where ( ) = + ( − 1) ( − 1), = and and does not exist.
,
10

6
0
2 2
2
( , ) is the Gaussian PDF with mean and variance b) If is a prime number and is an integer, there

2
. Taking expectation on both sides of Eq. 6, we get exists a with elements.
c) If is a prime number and is an integer, then

is called the subfield of and is known as
( ))
1 − ( ,
an extension field of .

d) Every field element has at least a primitive
−1 (7)

∞ element , such that all the elements can be

( )) ]
= [∫ ∑ tanh ( ) ( , , ( ),2 , expressed as the power of (except 0).
2
−∞
=2
3
For example: If can be constructed using ( ) = +
8
where + 1 and we assume the primitive element of be .
8
[ ℎ ( )] = Then we can represent all the elements of by powers of
8
2 1 2 2 3
∞ evaluated modulo ( ) as = , = , = + 1,

∑ ∫ tanh ( ) ( , , 2 ) and ( ) = 1 −
7
6
2
5
4
2
2
−∞ 2 = + , = + + 1, = + 1, and = 1.

[tanh ( )] , and = ( , 2 ). Hence, if we consider the field with the non-zero
8
2
elements of the field reported earlier. Thus, we can write
7
2
2
We average mean , ( ) over all the CNs degree to get the − 1 = ( − 1)( − )( − )( − − 1)( − −
2
2
final equation of the mean as )( − − − 1)( − − 1).
The emphasis of algebraic code is based on the construction
( )

of codes with a guaranteed minimum distance and

applying the algebraic structure of the codes to design
= ∑ −1 (1 bounded-distance decoding where complexity grows only

as a small power of . When there is a codeword of
−1 (8) length n and message of length k, the follows the

∞
singleton bound, which is given as
( )) ] )
− [∫ ∑ ℎ ( ) ( , , ( ), 2 ,
2
−∞

< − + 1, (9)
When there are errors in the channel, the number of errors
that can be corrected is
These are the essential steps in deriving threshold, i.e.,
signal to noise ratio (SNR) where the error rate reduces to a
very small number and determining degree distribution for
≤ ⌊ − 1⌋ 2 . (10)
⁄
the LDPC codes.
As we get a received word, it is decoded to a codeword
2.3 RS Codes
which is within of it. If there is no such code word,
then we declare a decoding failure. Otherwise, a decoding

– 292 –