BCH Codes

Information Theory &

Coding

OUTLINE

INTRODUCTION

BLOCK DIAGRAM

EXAMPLE

MERITS

DEMERITS

APPLICATIONS

•Powerful random error-correcting cyclic codes.

•Makes block size(n) smallest for given message

block(k) to obtain desired hamming distance.

•Discovered by Hocquenghem in 1959 and

independently by Bose and Chaudhuri in 1960.

•Most important subclass Reed-Solomon (RS)

codes.

•Berlekamp’s iterative algorithm and Chien’s search

algorithm—most efficient decoding algorithms.

•In technical terms a BCH code is a

multilevel cyclic variable-length digital error-correcting

code used to correct multiple random error patterns.

INTRODUCTION

INTRODUCTION

For positive pair of integers m≥3 and t, a (n, k) BCHcode has parameters:

Block length: n = 2m – 1

Number of check bits: n – k ≤ mt

Minimum distance: dmin ≥ 2t + 1

t<(2m – 1)/2 random errors detected and corrected.

So also called ‘t-error correcting BCH code’.

Major advantage is flexibility for block length andcode rate.

Generator polynomial specified in terms of its

roots from Galois Field GF(2k).

g(x) has α,α2,…, α2t and their conjugates as its roots.

We choose g(x) from xn + 1 polynomial factors by

taking xn-k as highest term.

INTRODUCTION

The parameters of some useful BCH codes

are:n k t Generator Polynomial

7 4 1 1 011

15 11 1 10 011

15 7 2 111 010 001

15 5 3 10 100 110 111

31 26 1 100 101

31 21 2 11 101 101 001

31 16 3 1 000 111 110 101 111

31 11 5 101 100 010 011 011 010 101

31 6 7 11 001 011 011 110 101 000

100 111

INTRODUCTION

BLOCK DIAGRAM

Figure-1: Block diagram of (15,7) BCH Encoder

BCH Encoder

(15, 7) BCH Encoder.

The 7 message bits (M0, M1….M6) are applied to the

parallel to serial shift register.

The output of parallel to serial shift register will be

sent to (15, 7) BCH Encoder module.

Using these message bits, parity bits are computed

and sent to serial to parallel shift register.

Then parity bits are appended to original message bits

to obtain 15 bit encoded data.

This entire encoding process requires 15 clock cycles.

Figure-2: Block diagram for (15, 7) BCH

Decoder.

BLOCK DIAGRAM

BCH Decoder

(15, 7) BCH decoder.

The decoding algorithm for BCH codes consists of

three major steps.

Calculate the syndrome value Si, i=1,2,….,2t from

the received word r(x).

Determine the error location polynomial s(x)

Find the roots of s(x) and then correct the errors

EXAMPLE

For a (31,21,2) BCH code:

Encoder: t = 2

g(x)=xⁿ+1=x³¹+1

=(x+1)(x¹⁰+x⁹+x⁸+x⁶+x⁵+x³+1)

(x²⁰+x¹⁷+x¹⁶+x¹³+x¹¹+x⁷+x⁶+x⁵+x²+x+1)

Here highest order term for g(x) must be chosen as

xⁿ⁻ᵏ=x³¹⁻²¹=x¹⁰

So g(x)= (x¹⁰+x⁹+x⁸+x⁶+x⁵+x³+1)

Message D: (0110011)

Data: d(x)=x⁵+x⁴+x+1

So code C(x)=d(x).g(x)

= (x⁵+x⁴+x+1)(x¹⁰+x⁹+x⁸+x⁶+x⁵+x³+1)

= x¹⁵+2x¹⁴+2x¹³+x¹²+2x¹¹+4x¹⁰+3x⁹+2x⁸

+2x⁷+2x⁶+2x⁵+2x⁴+x³+x+1

= x¹⁵+x¹²+ x⁹+ x³+x+1

• Codeword, C: (1001001000001011)

MERITS The principal advantage is the ease with which they can be decoded

using ‘syndrome decoding’ method.

Allows very simple electronic hardware to perform the task,

obviating the need for a computer, and meaning that a decoding

device may be made small and low-powered.

Highly flexible, allowing control over block length and acceptable

error thresholds, meaning that a custom code can be designed to a

given specification

Reed–Solomon codes, which are BCH codes, are used in

applications such as satellite communications, compact

disc players, DVDs, disk drives, and two-dimensional bar codes.

BCH codes are also useful in theoretical computer science.

Low amount of redundancy

Easy to implement in hardware

Widely used

DEMERITS

Complexity

Iterative and complex decoding algorithm

Decoder cannot decide whether a decoded package

is false or not.

APPLICATIONS

BCH Codes as Industry Standards:

(511, 493) BCH code in ITU-T. Rec. H.261

“video codec for audiovisual service at kb/s”

a video coding standard used for video

conferencing and video phone.

n=511 m=9

k=493

n-k=18 t=2

(40, 32) BCH code in ATM (Asynchronous

Transfer Mode) pp. 223-227.

is a shortened cyclic code that can correct

1-bit error and detect 2-bit errors.

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