Griesmer bound

In the mathematics of coding theory, the Griesmer bound, named after James Hugo Griesmer, is a bound on the length of binary codes of dimension k and minimum distance d. There is also a very similar version for non-binary codes.

Statement of the bound

For a binary linear code, the Griesmer bound says:

$n\geq \sum_{i=0}^{k-1} \left\lceil\frac{d}{2^i}\right\rceil.$

Proof

Let $N(k,d)$ denote the minimum length of a binary code of dimension k and distance d. Let C be such a code. We want to show that $N(k,d)\geq \sum_{i=0}^{k-1} \left\lceil\frac{d}{2^i}\right\rceil$.

Let G be a generator matrix of C. We can always suppose that the first row of G is of the form r = (1, ..., 1, 0, ..., 0) with weight d.

$G= \begin{bmatrix} 1 & \dots & 1 & 0 & \dots & 0 \\ \ast & \ast & \ast & & G' & \\ \end{bmatrix}$

The matrix G' generates a code C', which is called the residual code of C. C' has obviously dimension $k'=k-1$ and length $n'=N(k,d)-d$. C' has a distance d', but we don't know it. Let $u\in C^{\prime}$ s.t. $w(u)=d'$. There exists a vector $v\in (F_2)^d$ s.t. the concatenation $(v|u)\in C$. Then $w(v)+w(u)=w(v|u)\geq d$. On the other hand, also $(v|u)+r\in C$, since $r\in C$ and $C$ is linear, so $w((v|u)+r)\geq d$. But

$w((v|u)+r)=w(((1,1,...,1)+v)|u)=d-w(v)+w(u)$,

so this becomes $d-w(v)+w(u)\geq d$. By summing this with $w(v)+w(u)\geq d$, we obtain $d+2w(u)\geq 2d$. But $w(u)=d'$, so we get $d'\geq d/2$. This implies $n'\geq N(k-1,d/2)$, therefore $n'\geq \left\lceil N(k-1,d/2)\right\rceil$ (due to the integrality of n'), so that $N(k,d)\geq \left\lceil N(k-1,d/2)\right\rceil +d$. By induction over k we will eventually get $N(k,d)\geq \sum_{i=0}^{k-1} \left\lceil\frac{d}{2^i}\right\rceil$ (note that at any step the dimension decreases by 1 and the distance is halved, and we use the identity $\left\lceil\frac{\left\lceil a/2^{k-1}\right\rceil}{2}\right\rceil = \left\lceil\frac{a}{2^k}\right\rceil$ for any integer a and positive integer k).

The bound for the general case

For a linear code over $\mathbb{F}_q$, the Griesmer bound becomes:

$n\geq \sum_{i=0}^{k-1} \left\lceil\frac{d}{q^i}\right\rceil.$

The proof is similar to the binary case and so it is omitted.

See also

• Singleton bound
• Hamming bound
• Gilbert-Varshamov bound
• Johnson bound
• Plotkin bound
• Elias Bassalygo bound

References

• J. H. Griesmer, "A bound for error-correcting codes," IBM Journal of Res. and Dev., vol. 4, no. 5, pp. 532-542, 1960.