An Efficient Interpolation-Based Chase BCH Decoder

IJSRD - International Journal for Scientific Research & Development| Vol. 2, Issue 08, 2014 | ISSN (online): 2321-0613
All rights reserved by www.ijsrd.com 81
An Efficient Interpolation-Based Chase BCH Decoder
D. Prabhakar1
B. Rajesh2
B. Shirisha3
1
M.Tech Student 2,3
Assistant Professor
1,2,3
Department of Electronics and Communication Engineering
1,2,3
Vathsalya Institute of science and technology, R.R Dist,T.S, India
Abstract— Error correction codes are the codes used to
correct the errors occurred during the transmission of the
data in the unreliable communication mediums. The idea
behind these codes is to add redundancy bits to the data
being transmitted so that even if some errors occur due to
noise in the channel, the data can be correctly received at the
destination end. Bose,Ray Chaudhuri, Hocquenghem
(BCH)codes are one of the error correcting codes. The BCH
decoder consists of four blocks namely syndrome block,
chien search block and error correction block. This paper
describes a new method for error detection in syndrome and
chien search block of BCH decoder. The proposed
syndrome block is used to reduce the number of
computation by calculating the even number syndromes
from the corresponding odd number syndromes.
Keywords: BCH Codes, Syndrome Block, Chien search
Block, Error detection
I. INTRODUCTION
BCH CODES can be found in manyapplications, including
flash memory, optical transport network, and digital video
broadcasting. For a BCH code with minimum distance dmin,
traditional hard-decision decoding (HDD) algorithms, such
as the Berlekamp’s algorithm, can correct t = dmin/2 errors.
Through flipping the η least reliable bits and trying 2η
test
vectors,the soft-decision Chase algorithm can correct up to t
+ η errors. It also has better error-correcting performance
than the generalized minimum distance (GMD) decoder and
the soft-decision decoder in [1], which assumes that all but
one error are located in the 2t least reliable bits. To reduce
the complexity of the Chase BCH decoding, one-pass
schemes have been proposed to derive the error locators for
all test vectors in one run based on the Berlekamp’s
algorithm [2], [3]. Comparatively, the scheme in [2] has
lower complexity.
This brief proposes a novel interpolation-based
Chase BCH decoder. It is inspired by the interpolation-based
Chase decoder for Reed–Solomon (RS) codes. Nevertheless,
by making use of the binary property of BCH codes,
substantial modifications and simplifications are developed
in this brief. In particular, instead of employing expensive
parallel Chien search, the selec-tion of the interpolation
output leading to successful decoding is achieved by simple
evaluation value computation without any error-correcting
performance or code rate loss. In addition, the recovery of
each code word bit is done through testing the evaluation
values of two low-degree polynomials. From architectural
analysis, the proposed decoder with η = 4 for an example
(4200, 4096) BCH code has 2.3 times higher efficiency in
terms of throughput-over-area ratio than the Chase decoder
based on the Berlekamp’s algorithm [4], while achieving the
same error-correcting performance.
The structure of this brief is as follows. Section II in-
troduces the Chase decoding. The interpolation-based Chase
BCH decoding and its implementation architectures are pro-
posed in Sections III and IV, respectively. Section V
provides the hardware complexity analysis, and conclusions
are drawn in Section VI
II. CHASE DECODING
The fundamental difference between a ST Chase decoder
and previously published decoders is that the ST Chase
decoder concentrates on correcting bit errors instead of
symbol errors. In this section, we will combineideas from
sphere decoding and binary block decoding to create a low
complexity MIMO decoder.
III. INTERPOLATION-BASED CHASE BCH DECODER
An (n, k) t-bit error-correcting BCH code over GF (2p
) is a
subfield subcode of an (n k)
t-symbol error-correcting RS
code over GF (2p
). In another word, all the (n, k) BCH code
words form a subset of the (n, k) RS code words. n_
k= 2t,
and n − k is equal to or slightly less than pt. The
interpolation-based decoding is developed based on the
interpretation that the code word symbols are evaluation
values of the message poly-nomial. BCH codes cannot be
encoded this way since the eval-uation values of a binary
message polynomial over finite field elements are usually
not binary. Hence, BCH code words are considered as RS
code words in order to apply the interpolation-based
decoding. Applying RS systematic re-encoding to the last k_
code positions, n – k points remain to be interpolated for
each test vector. The same backward–forward interpolation
scheme [7], [8] can be adopted to derive the interpolation
results of all vectors in one run. Nevertheless, by making
use of the property that r is binary in BCH decoding,
substantial simplifications can be made on the polynomial
selection and code recovery section.
High level decoder design
The BCH decoder has four modules as mentioned
below:
 Syndrome Calculator
 Solving the key equation
 Error Location
 Error Correction
IV. SYNDROME CALCULATOR
The syndrome calculator is the first module at the decoder
also, the design of this module is almost same for all the
BCH code decoder architecture. The input to this module is

(IJSRD/Vol. 2/Issue 08/2014/021)
corrupted codeword. The equations for the codeword,
received bits and the error bits are given in equations (5.1),
(5.2), (5.3)[4].
Codeword equation
c(x) = c0 + c1x + c2x2
+ ... + cn-1xn-1
Received bits equation
r(x) = r0 + r1x + r2x2
+ ... + rn-1xn-1
Error bits equation
e(x) = e0 + e1x + e2x2
+ ... + en-1xn-1
Thus, the final transmitted data polynomial
equation is given as below:
r(x) = c(x) + e(x)
The 1st
step at the decoding process is to store the
transmitted data polynomial in the buffer register and then to
calculate the syndromes sj. The important characteristic of
the syndromes is that depends on only error location not on
transmitted information. The equation of the syndromes are
given as follows [4]:
Define the syndromes Sj as
S rj i
i j
i
n
 


 
0
1
for (1  j  2t). Since rj = cj + ej
(j = 0, 1, ...., n-1)
Rewrite the syndrome equation as:
S c e c ej i i
i j
i
n
i
i j
i
i j
i
n
i
n
   


 




 ( )  
0
1
0
1
0
1
By the definition of BCH codes
ci
i j
i
n
 


 0
0
1
for (1  j  2t)
Thus,





1
0
n
i
ji
ij eS 
The above equation is indicates the output of the
syndrome calculator. From the equation it can be observed
that the syndromes are depends on only error polynomial
e(x), so if there is no error occurs during the transmission
then all the generated syndromes will be zero.
Architecture For Proposed Polynomial Selection
Conventional Syndrome Calculator
V. KEY EQUATION SOLVER
The second stage in the decoding process is to find the co-
efficient of the error location polynomial using the
generated syndromes in the previous stage. The error
location polynomial is given as: (x) = 0 + 1x + ... + txt.
The relation between the syndromes and the error location
polynomial is given as below [4]:
St i j
j
t
j 

 
0
0 (i= 1, ..., t)
There are various algorithms used to solve the key
equation solver. This project is using the Inversion less
Berlekamp Massey algorithm to solve the key equation.
Berlekamp Massey Algorithm.
The steps of berlekamp Massey algorithm is given as below:
(1) First step is to calculate error syndromes Sj.
(2) Initialize the k = 0, Λ(0)
(x) = 1, L = 0 and T(x) = x
(3) Assign k = k + 1 and then the discrepancy Δ(k)
is
then calculated as follows:
(4) If the value of Δ(k)
, 2L ≥ equals 0, then go-to step 7.
(5) Calculate the Λ(k)
(x) = Λ(k-1)
(x) - Δ(k)
T(x)
(6) Set the value of L = k – L and T(x) is calculated as
T(x) = Λ(k-1)
(x) / Δ(k)
(7) Set T(x) = x.T(x).
(8) If the value of k< 2t , then go-to step 3
(9) Continue for i = 2t – 1 and then End.
The decoder of this project is based on the
Inversion-less Berkelamp algorithm (iBM) for Key
Equation Calculation. The architecture for iBM
algorithm is explained in detail in the next chapter.
VI. ERROR LOCATION – CHAIN’S SEARCH
To calculate the error location is the next step of decoding
process, which can be done using chain search block.
A. Chain Search Algorithm
The roots are calculated as follows [12] [4]:
(1) For each power of α for ( j = 0 to n – 1), αj
is taken
as the test root
(2) Calculate the polynomial coefficients, of the
current root using, coefficients of the past iteration,
using, Λ i
(j)
= Λ i
(j-1)
αi
during the jth
iteration
(3) Calculate the sum of the polynomial coefficients
(4) The sum is equal to
(5) Continue to Step 1 till j = n-1

(IJSRD/Vol. 2/Issue 08/2014/021)
Chain’s Search architecture – Error Location
VII. ERROR CORRECTION
The output of the chain search block is called roots of
equation. The reciprocal of the roots of equations are added
with the corresponding location of the corrupted codeword
received by decoder. The result of this addition is the
original codeword that was encoded by the encoder before
transmission.
CONCLUSION
This brief developed an efficient interpolation-based
onepass Chase BCH decoder. By making use of the binary
property of BCH codes, novel polynomial selection and
code word recovery schemes were proposed. In particular,
the proposed polynomial selection led to significant
complexity reduction without sacrificing the error-correcting
performance. Future work will be directed to further
speeding up the interpolationbased BCH decoder without
incurring large area overhead.
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An Efficient Interpolation-Based Chase BCH Decoder

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