# Unique decoding Radius in Reed Solomon Codes

In one of the coding theory books I read the unique decoding radius for Reed Solomon codes is $$\frac{1-\rho}{2}$$. Precisely, if the relative distance be less than these amount so the receiver is able to reconstruct the noisy received code-word uniquely. While for bigger amount of relative distance there are other code-words that are closer to the received code-word instead of the original code-word.

How this bound is calculated?

## 1 Answer

Let the linear block code $$C$$ with the minimum distance of $$d_{min}$$ which this amount is either odd or even so for each integer $$t$$ we can define the below inequality: $$$$2t+1 \leq d_{min} \leq 2t+2$$$$ Suppose $$W$$ is all code-words in code $$C$$ except $$V$$ which is the original code-word along with the channel and $$r$$ is the received code-word. Therefore, the below inequality according to one of the famous Lemma is satisfied. $$$$d(V,r)+d(r,W) \geq d(V,W)$$$$ Since $$V$$, $$W$$ $$\in$$ $$C$$, $$$$d(V,W) \geq d_{min} \geq 2t+1$$$$ Now, suppose the number of error between $$V$$ and $$r$$ is $$l$$ i.e. $$d(V,r)=l$$. So $$$$d(r,w) \geq 2t+1-l$$$$ If $$l$$ $$\leq$$ $$t$$ then $$$$d(r,w) \geq t$$$$

In other words, if the number of errors between the transmitted and the received code-words be less than $$t$$ so the code-word $$V$$ is the closest code-word to $$r$$ than all other code-words in code $$C$$. i.e. $$$$d(V,r) < \frac{d_{min}-1}{2}$$$$

We can generalize this bound for all linear block codes such as Reed Solomon codes. The $$d_{min}$$ for RS codes is $$n-k+1$$ so we can conclude: $$$$d(V,r) < \frac{n-k+1-1}{2}=n\frac{1-\frac{k}{n}}{2}$$$$ So $$$$\delta < \frac{1-\rho}{2}$$$$

It is exactly the bound for the RS codes to obtain uniquely the original code-word.