Gilbert–Varshamov bound

In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert[1] and independently Rom Varshamov[2]) is a limit on the parameters of a (not necessarily linear) code. It is occasionally known as the Gilbert–Shannon–Varshamov bound (or the GSV bound), but the name "Gilbert–Varshamov bound" is by far the most popular. Varshamov proved this bound by using the probabilistic method for linear code. For more about that proof, see: GV-linear-code.

Statement of the bound

Let

A_q(n,d)

denote the maximum possible size of a q-ary code C with length n and minimum Hamming weight d (a q-ary code is a code over the field \mathbb{F}_q of q elements).

Then:

A_q(n,d) \geq \frac{q^n}{\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j}.

Proof

Let C be a code of length n and minimum Hamming distance d having maximal size:

|C|=A_q(n,d).\,

Then for all x\in\mathbb{F}_q^n , there exists at least one codeword c_x \in C such that the Hamming distance d(x,c_x) between x and c_x satisfies

d(x,c_x)\leq d-1

since otherwise we could add x to the code whilst maintaining the code's minimum Hamming distance d – a contradiction on the maximality of |C|.

Hence the whole of \mathbb{F}_q^n is contained in the union of all balls of radius d − 1 having their centre at some c \in C :

\mathbb{F}_q^n =\cup_{c \in C} B(c,d-1).\,

Now each ball has size

\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j

since we may allow (or choose) up to d-1 of the n components of a codeword to deviate (from the value of the corresponding component of the ball's centre) to one of (q-1) possible other values (recall: the code is q-ary: it takes values in \mathbb{F}_q^n). Hence we deduce

\begin{align} |\mathbb{F}_q^n| & = |\cup_{c \in C} B(c,d-1)| \\ \\ & \leq \sum_{c \in C} |B(c,d-1)| \\ \\ & = |C|\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j \\ \\ \end{align}

That is:

A_q(n,d) \geq \frac{q^n}{\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j}

(using the fact: |\mathbb{F}_q^n|=q^n).

An improvement in the prime power case

For q a prime power, one can improve the bound to A_q(n,d)\ge q^k where k is the greatest integer for which

q^k < \frac{q^n}{\sum_{j=0}^{d-2} \binom{n-1}{j}(q-1)^j}.

See also

References

  1. ↑ Gilbert, E. N. (1952), "A comparison of signalling alphabets", Bell System Technical Journal 31: 504–522, doi:10.1002/j.1538-7305.1952.tb01393.x.
  2. ↑ Varshamov, R. R. (1957), "Estimate of the number of signals in error correcting codes", Dokl. Acad. Nauk SSSR 117: 739–741.