Griesmer Bound - Proof

Proof

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.

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 and length . C' has a distance d', but we don't know it. Let s.t. . There exists a vector s.t. the concatenation . Then . On the other hand, also, since and is linear, so . But

,

so this becomes . By summing this with, we obtain . But, so we get . This implies, therefore (due to the integrality of n'), 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).

Read more about this topic:  Griesmer Bound

Famous quotes containing the word proof:

    “It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
    —William Shakespeare (1564–1616)

    “The insatiable thirst for everything which lies beyond, and which life reveals, is the most living proof of our immortality.”
    —Charles Baudelaire (1821–1867)

    “There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
    —François, Duc De La Rochefoucauld (1613–1680)