linear code correcting and detecting capability

[neoflash]

There is equation from Skalar book saying that

dmin ≥ α + β + 1; (page 346)

α is correcting capability and β is detecting capability.

There is trade off between α and β since their sum is contant.

My question is that why there is this kind of trade off?

Linear coding should be able to detect all e (=dmin-1) error patterns, which is defined in page 345. Is not these two equations contradicting?

 

[ilya79]

>>My question is that why there is this kind of trade off?

I suppose the answer in maximum likelihood decoding

For example let’s take 31,26 BCH code- its dmin is 3.

(n,k) –code with n-31 k-26

c0,c1 are code words
+-vector in n-dimensional space

+ c0
|
|
+ (c0+err) or (c1+2err)
|
|
+ (c0+2*errr) or (c1+err)
|
|
+ c1

If you are only detecting errors than you are capable of detecting 2 errors. If you are correcting and detecting errors than in the case of transmitting c0 and 2 errors happened, ML decoder will correct it to code word c1(because received vector is clause to c1 then to c0) and will make 3 error’s.

 

[samuraign]

Hi,
Do u mean maximum likelihood decoding is the best decoding
algo for BCH.
Could you suggesr best algo for decoding 2errors in BCH code.

 

[m_llaa]

Hi
correction and detection are 2 capability in channel error correcting codes.
In some applications reciever just want to detect errors not correct them. in this case codes with small correction capability and large detection capability are used.
I suggest u "Error correcting codes" book by Shu Lin
in this book this topic explained breifly in intruduction or chapter 1 i think.