Synthesis-by-analysis of BCH Codes

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3 years ago

Abstract - in this paper we propose techniques to be blindly

Synthesize the Polinomial Generator BCH code. The proposed

The technique involves finding the largest general divider (GCD)

Among the different codewords and block lengths. Based on this

Calculation of gcd combinatorial, correlation value found. For

Valid block length, the results of the gcd calculation repeatedly too

into the polinomial generator or multiple multiples of the higher sequence.

This high-order polynomial is a factor under the modulo-2

the operation, and one of the factors generated always a generator

Polynomial which further increases correlation value. That

The correlation plot produced for different polynomials shows very

High value for the correct length of the block and generator that is valid polyno. mial. Know the length of the valid block and polinomial generator,

All other parameters include the number of digits checking parity (n n n k), minimum distance and the ability to correct errors t already exposed.

GCD index requirements, blind estimation, polynomial generator,

Correlation value.

I. Introduction

Error control is mandatory to fight inevitably

Random and broken errors in digital communication channels.

There are various error control codes among which BosereChaudhuri-Hocquenghem (BCH) Cyclic codes are very famous

and widely used in digital communication channels. This

The code is marked with a block length

N, the number of Pary�Check Digit (NNK) and Dmin minimum distance. generator

Polynomial BCH code is determined as Mul� Tiple (LCM) is at least common (LCM) of the minimal polynomial (x) where 1 ≤ I ≤ 2T, T is a code correcting error.

In the matter of eavesdropping on communication channels,

there is no previous knowledge available except grand

Bitstream. Source information is packed into numbers

various layers before sending it to communication

channel. In this scenario, someone must estimate blindly differently

Parameters in each layer. Very few papers dealing with

Synthesis problems and reconstruction of error control code

from eavesdropping bitstreams. Rice [1] Presents Engineering

To estimate tariff parameters

Convolutional code 1 / n

which is generalized by Philiol [2] for other rates too

for convolutional code punctured. Burel [3] suggest blind

Estimated Characteristics of Encoder and Interleaver based

Linear algebraic theory. Barbier [4] analyzes various techniques

To blindly restore the parameters of the turbo code encoder. In

2006, Cluzeau [5] introduced the Algorithm Gallager version

m.

with a weighted parity examination equation to recover LDPC and

Other block code.

In this paper, the synthesis-by-analysis of BCH code has been determined before. The proposed technique focuses on parameters

Estimation in the channel coding layer in general and on BCH

Code in particular. In our work, key parameters become

It is estimated that a polynomial generator for valid block length n. Know the length of the N block and the G (X) polynomial generator, all other parameters can be easily found and BCH

code can be translated without the previous knowledge of

transmission side.

We assume that we have access to steam BCH

Bitstream encoded. This assumption is simulated by producing

Vector test for various BCH codes

(n, k, t

). For specifics

(n, k, t

) Code, the test vector is forwarded to the proposed

Algorithm and gcd found for two codewords at first

repetition. The algorithm then steps through various available

codewords in a combinatorial way. For each combination

codewords, the GCD value used to find a correlation for

Different polynomial candidates. For a valid block length

D.

The correct polynomial generator, this correlation piled up

Very high value. For some generator codewords

Polynomial is not revealed, but by repeating detected

Polynomial operation under the modulo-2, the desired general

R.

Polynomial is taken and the correlation value increases feathers. After planning the correlation value, the desired generator

Polynomials for valid block lengths are exposed to very explicit.

The proposed technique exploits cyclic relationships

between codewords BCH code. This technique works

Perfect for bitstream without sound. However, it's equally valid

If there are certain errors in some codewords. Since this

is an analytical technique unlike real-time decoding, the effect

Noisy codewords can be reduced by increasing the number

Codewords. Correlation values ​​accumulated to increase

Number of test vectors with a very slight increase in processing

And therefore the algorithm works for bitstream which is noisy too.

This paper is compiled as follows. In Part II, we remember

the principles of construction of the BCH code along with standards

Procedure for generating test vectors. Part III provides refreshment

about GCD and Euclid algorithm follow detection

Polynomial mathematics generator in Section IV. Simulation.

N.

The results are displayed in the V. section.

II. BCH CONSTRUCTION

Given any positive integer(m≥ 3) and error-correcting capability(t <2mm1), a BCH code can be generated with

The following parameters:-

Block Length:n = 2mm1,

Minimum distance: dmin ≥ 2t + 1,

Number of parity-check digits: n n k ≤ mt,

The generator polynomial g(X) is the LCM of

φ1(X), φ2(X), · · ·, φ2t(X) :

g(X) = LCM{φ1(X), φ2(X), · · ·, φ2t(X)} (1)

Since every even power of primitive element, α has the same

minimal polynomial as some preceding odd power of α, hence

(1) can be reduced to:-

g(X) = LCM{φ1(X), φ3(X), · · ·, φ2tt1(X)} (2)

Test vectors (encoding in systematic form) are generated by

following the standard encoding steps [6] which are:-

1) Pre-multiply, k information digits, message polynomial

with Xnnk

i.e. Xnn ku(X).

2) Calculate parity check polynomial b(X) from dividing

Xnn ku(X) by g(X).

3) Append b(X) with Xnn ku(X) to obtain the code poly￾nomial v(X) = b(X) + Xnn ku(X).

The above steps can be realized by a division circuit based

on linear (nn k) stage shift register with feedback connections

based on g(X)

The operation of the coding circuit [6] is described as

following:

1) Initially, the gate was turned on. Information digit, K,

Message of polynomial u (x) = U0 + U1X + ··· + ukk1xkk 1

fed to the circuit and transmitted into

channel. Feed digit information K into the circuit

Equivalent to Duplicate U (X) by XNN K

, When

All digit information is shifted into the circuit,

(n n k) digit in the register form the rest.

2) the gate is then turned off, since the register now

contains digits check parity (n n n n n k).

3) voters are changed to the right position to send parity

Check the digit to the channel. This (nn k) parity check

digit along with k digit information forming cyclic

Codeword in a systematic form.

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