Johnson bound

In applied mathematics, the Johnson bound (named after Selmer Martin Johnson) is a limit on the size of error-correcting codes, as used in coding theory for data transmission or communications.

Definition
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.


 * $$d = \min_{x,y \in C, x \neq y} d(x,y),$$

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.

Denote by $$|C|$$ the number of elements in $$C$$. Then, we define $$A_q(n,d)$$ to be the largest size of a code with length $$n$$ and minimum distance $$d$$:


 * $$ A_q(n,d) = \max_{C \in C_q(n,d)} |C|.$$

Similarly, we define $$A_q(n,d,w)$$ to be the largest size of a code in $$C_q(n,d,w)$$:


 * $$ A_q(n,d,w) = \max_{C \in C_q(n,d,w)} |C|.$$

Theorem 1 (Johnson bound for $$A_q(n,d)$$):

If $$d=2t+1$$,


 * $$ A_q(n,d) \leq \frac{q^n}{\sum_{i=0}^t {n \choose i} (q-1)^i + \frac{{n \choose t+1} (q-1)^{t+1} - {d \choose t} A_q(n,d,d)}{A_q(n,d,t+1)} }. $$

If $$d=2t+2$$,


 * $$ A_q(n,d) \leq \frac{q^n}{\sum_{i=0}^t {n \choose i} (q-1)^i + \frac{{n \choose t+1} (q-1)^{t+1} }{A_q(n,d,t+1)} }. $$

 Theorem 2 (Johnson bound for $$A_q(n,d,w)$$):

(i) If $$d > 2w,$$


 * $$ A_q(n,d,w) = 1. $$

(ii) If $$d \leq 2w$$, then define the variable $$e$$ as follows. If $$d$$ is even, then define $$e$$ through the relation $$d=2e$$; if $$d$$ is odd, define $$e$$ through the relation $$d = 2e - 1$$. Let $$q^* = q - 1$$. Then,


 * $$ A_q(n,d,w) \leq \left\lfloor \frac{n q^*}{w} \left\lfloor \frac{(n-1)q^*}{w-1} \left\lfloor \cdots \left\lfloor \frac{(n-w+e)q^*}{e} \right\rfloor \cdots \right\rfloor \right\rfloor \right\rfloor $$

where $$\lfloor \rfloor$$ is the floor function.

Remark: Plugging the bound of Theorem 2 into the bound of Theorem 1 produces a numerical upper bound on $$A_q(n,d)$$.