In coding theory, the bound of parameters such as rate R, relative distance, block length, etc. is usually concerned. Here Gilbert–Varshamov bound theorem claims the lower bound of the rate of the general code. Gilbert–Varshamov bound is the best in term of relative distance for codes over alphabets of size less than 49.
Gilbert–Varshamov bound theorem
Theorem: Let . For every , and , there exists a code with rate , and relative distance .
Here is the q-ary entropy function defined as follows:
The above result was proved by Edgar Gilbert for general code using the greedy method as here. For linear code, Varshamov proved using the probabilistic method for the random linear code. This proof will be shown in the following part.
To show the existence of the linear code that satisfies those constraints, the probabilistic method is used to construct the random linear code. Specifically the linear code is chosen randomly by choosing the random generator matrix in which the element is chosen uniformly over the field . Also the Hamming distance of the linear code is equal to the minimum weight of the codeword. So to prove that the linear code generated by has Hamming distance , we will show that for any . To prove that, we prove the opposite one; that is, the probability that the linear code generated by has the Hamming distance less than is exponentially small in . Then by probabilistic method, there exists the linear code satisfying the theorem.
By using the probabilistic method, to show that there exists a linear code that has a Hamming distance greater than , we will show that the probability that the random linear code having the distance less than is exponentially small in .
We know that the linear code is defined using the generator matrix. So we use the "random generator matrix" as a mean to describe the randomness of the linear code. So a random generator matrix of size contains elements which are chosen independently and uniformly over the field .
Recall that in a linear code, the distance = the minimum weight of the non-zero codeword. This fact is one of the properties of linear code.
Denote be the weight of the codeword . So
Also if codeword belongs to a linear code generated by , then for some vector .
By Boole's inequality, we have:
Now for a given message , we want to compute
Denote be a Hamming distance of two messages and
Then for any message , we have: .
Using this fact, we can come up with the following equality:
Due to the randomness of , is a uniformly random vector from .
Let is a volume of Hamming ball with the radius . Then:
(The later inequality comes from the upper bound of the Volume of Hamming ball)
By choosing , the above inequality becomes
Finally , which is exponentially small in n, that is what we want before. Then by the probabilistic method, there exists a linear code with relative distance and rate at least , which completes the proof.
- The Varshamov construction above is not explicit; that is, it does not specify the deterministic method to construct the linear code that satisfies the Gilbert–Varshamov bound. The naive way that we can do is to go over all the generator matrices of size over the field and check if that linear code has the satisfied Hamming distance. That leads to the exponential time algorithm to implement it.
- We also have a Las Vegas construction that takes a random linear code and checks if this code has good Hamming distance. Nevertheless, this construction has the exponential running time.
- Gilbert–Varshamov bound due to Gilbert construction for the general code
- Hamming Bound
- Probabilistic method