Page 337 - Kaleidoscope Academic Conference Proceedings 2024

Innovation and Digital Transformation for a Sustainable World

error occurs if we decode the received codeword to the the received polynomial in Section 2, Eq. 14. Thus, we
wrong code. Correct decoding can only happen if the write the equation of syndrome as
received word is within the distance of the transmitted
codeword. ( ) = ( ) + ( ) = ( ). (16)

2.3.1 Code construction Hence, for the block length , we get syndromes derived as
Now, to find the generator polynomials for cyclic codes of −1

block length , we have to first factorize − 1 as the = ∑ , (17)

product of its prime factors. =0
where = 2 . Suppose has errors at locations , , …

1

− 1 = ( ) ( ) ( ) … . . ( ) (11) with corresponding error values at the location ≠ 0. Let

3
1
2

= we get
Any combination of these factors can be multiplied together
to form a generator polynomial ( ). If the prime factors

are distinct, then we obtain − different non-trivial = ∑ ; = 1,2, … ,2 (18)

cyclic codes of block length . The two trivial cases that are
=0

being disregarded as ( ) = 1 and ( ) = − 1. As we

already know, ( ) is a factor of − 1. Thus, the ( ) of
If we know then we know the location of errors. If
the RS codes can be written as 4
= ; then an error is in the received bit 4. Hence is

1
known as error locators. From Eq. (18), we obtain the 2
( ) = [ ( ), ( ), … ( )] (12) equalities that relate the error locations of ( ) and the 2

2
1
syndrome components, for 1 ≤ ≤ 2 . For example,

where ( ), ( ), … ( ) are the minimal polynomials of = + + ⋯ + , = + + ⋯ + 2 or
2
2

1
2
1
1
2
2
1
2
the zeroes of ( ). Let ( ) be the codeword polynomial. 2 = 2 = 1 2 + 2 2 + ⋯ + . This gives 2 equations
2

The encoding rule to generate the code words from ( ) is in the unknown error locations. Now, to avoid these non-
linear equations directly, we use the Berlekamp-Massey
( ) = ( ) ( ), (13) algorithm. We rewrite the syndrome equation using
Newton’s identity as
where ( ) is the information polynomial. The received
polynomial is represented as
= − ∑ Λ ; = + 1, + 2, … ,2 (19)
−
( ) = ( ) + ( ), (14) =0
where ( ) is an error polynomial. Now, RS codes are We then find the error locator polynomial after solving the
defined with respect to the roots of the ( ) in finite fields. above equation. Using the Chein search algorithm, the roots
of the error locator polynomial are found. We take the
2.3.2 Decoding inverse of those and correct the errors at those locations, i.e.,
flip the bit present at that received bit location. Hence,
The message bits are encoded using an RS encoder and decoding is performed for RS codes.
sent through the channel. The decoding of the RS code is
done in three steps. First, the syndrome is computed, and 3. PROPOSED SYSTEM MODEL
then the Berlekamp-Massey algorithm is used to generate
the error location polynomial from the syndrome In this section, we discuss the system model shown in Fig.
polynomial. The next step is to get error locations by 3 pertaining to the BICM precoded RS-coded system.
finding the roots of the error location polynomial. Errors Though the proposed system is scalable for any wireless
are corrected, and an estimated decoder information word is channel, but we do the simulation study for the optical
generated. An estimated sequence is obtained and frequency band at the wavelength of 1550 nm. We use a BP
compared with the transmitted sequence to locate error based soft-decision decoder for LDPC codes. We also
locations that the decoder cannot correct. Here, we discuss optimize the degree distribution for irregular LDPC (1912,
the general decoding procedure for the RS codes. We 956), owing to the slow fading channel characteristics at
consider the error polynomial as high-frequency communication scenarios. The optimized
2
degree distribution pairs are: ( ) = 0.4266 + 0.5734
3
and ( ) = 0.0003 + 0.9997 4 with the code’s
1
( ) = −1 −1 + −2 −2 + ⋯ + + (15) spectral efficiency of 0.5 which we optimize using our
1
0
previous work shown in [11]. We then interleave the coded
Here, ( ) is the channel noise-induced error pattern during bits through a bitwise and infinite (hypothetically, in reality,
the transmission of the code polynomial ( ). We represent its depth is equivalent to the size of codeword bits) depth

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