Griesmer bound

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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[edit]

For a binary linear code, the Griesmer bound is:

Proof[edit]

Let 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

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.

The matrix generates a code , which is called the residual code of has obviously dimension and length has a distance but we don't know it. Let be such that . There exists a vector such that the concatenation Then On the other hand, also since and is linear: But

so this becomes . By summing this with we obtain . But so we get This implies

therefore due to the integrality of

so that

By induction over k we will eventually get

Note that at any step the dimension decreases by 1 and the distance is halved, and we use the identity

for any integer a and positive integer k.

The bound for the general case[edit]

For a linear code over , the Griesmer bound becomes:

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

See also[edit]

References[edit]

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