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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:

$n\ge \sum _{i=0}^{k-1}\lceil \frac{d}{2^i}\rceil .$

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 $N(k,d)\ge \sum _{i=0}^{k-1}\lceil \frac{d}{2^i}\rceil$.

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.

$G=\left[\begin{array}{cccccc}1& \dots & 1& 0& \dots & 0\\ \ast & \ast & \ast & & G^{\prime }& \end{array}\right]$

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

$w((v|u)+r)=w(((1,1,\mathrm{...},1)+v)|u)=d-w(v)+w(u)$,

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

The bound for the general case

For a linear code over $F_q$, the Griesmer bound becomes:

$n\ge \sum _{i=0}^{k-1}\lceil \frac{d}{q^i}\rceil .$

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

References

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