Zyablov bound is a lower bound on the rate and relative distance of concatenated codes.
Statement of the bound
Let be the rate of the outer code and be the relative distance, then the rate of the concatenated codes satisfies the following bound.
where is the rate of the inner code .
Let be the outer code, be the inner code.
Consider meets the Singleton bound with rate of , i.e. has relative distance In order for to be an asymptotically good code, also needs to be an asymptotically good code which means, needs to have rate and relative distance .
Suppose meets the Gilbert-Varshamov bound with rate of and thus with relative distance
then has rate of and
Expressing as a function of
Then optimizing over the choice of r, we get that rate of the Concatenated error correction code satisfies,
This lower bound is called Zyablov bound (the bound of is necessary to ensure ). See Figure 2 for a plot of this bound.
Note that the Zyablov bound implies that for every , there exists a (concatenated) code with rate
We can construct a code that achieves the Zyablov bound in polynomial time. In particular, we can construct explicit asymptotically good code (over some alphabets) in polynomial time.
Linear Codes will help us complete the proof of the above statement since linear codes have polynomial representation. Let Cout be an Reed-Solomon error correction code where (evaluation points being with , then .
We need to construct the Inner code that lies on Gilbert-Varshamov bound. This can be done in two ways
- To perform an exhaustive search on all generator matrices until the required property is satisfied for . This is because Varshamovs bound states that there exists a linear code that lies on Gilbert-Varshamon bound which will take time. Using we get , which is upper bounded by , a quasi-polynomial time bound.
- To construct in time and use time overall. This can be achieved by using the method of conditional expectation on the proof that random linear code lies on the bound with high probability.
Thus we can construct a code that achieves the Zyablov bound in polynomial time.
References and External Links
- MIT Lecture Notes on Essential Coding Theory – Dr. Madhu Sudan
- University at Buffalo Lecture Notes on Coding Theory – Dr. Atri Rudra
- University of Washington Lecture Notes on Coding Theory- Dr. Venkatesan Guruswami