Johnson Bound

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Johnson Bound

Let

C

be a q-ary Code of length

n

, i.e. a subset of

\mathbb{F}_q^n

. Let

d

be the minimum distance of

C

, i.e.

:

d = \min_{x,y \in C, x 
eq y} d(x,y)

where

d(x,y)

is the Hamming Distance between

x

and

y

.

Let

C_q(n,d)

be the set of all q-ary codes with length

n

and minimum distance

d

and let

C_q(n,d,w)

denote the set of codes in

C_q(n,d)

such that every element has exactly

w

nonzero entries.

:<math> A Q(n,d)

\max_{C \in C_q(n,d)} C</math>

:<math> A Q(n,d,w)

\max_{C \in C_q(n,d,w)} C</math>