Johnson bound

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The Johnson bound is a bound on the size of error-correcting codes.

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 Cq(n,d) be the set of all q-ary codes with length n and minimum distance d and let Cq(n,d,w) denote the set of codes in Cq(n,d) such that every element has exactly w nonzero entries.

Denote by | C | the number of elements in C. Then, we define Aq(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 Aq(n,d,w) to be the largest size of a code in Cq(n,d,w):

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

Theorem 1 (Johnson bound for Aq(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,

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 Aq(n,d,w)):

(i) If d > 2w,

Aq(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 \lfloor \frac{n q^*}{w} \lfloor \frac{(n-1)q^*}{w-1} \lfloor \cdots \lfloor \frac{(n-w+e)q^*}{e} \rfloor \cdots \rfloor \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 Aq(n,d).

[edit] See also

[edit] References

  • S. Johnson, "A new upper bound for error correcting codes," IRE Transactions on Information Theory, pp. 203--207, April 1962.
  • W. C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003.
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