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 formula_1 be a "q"-ary code of length formula_2, i.e. a subset of formula_3. Let formula_4 be the minimum distance of formula_1, i.e. where formula_7 is the Hamming distance between formula_8 and formula_9. Let formula_10 be the set of all "q"-ary codes with length formula_2 and minimum distance formula_4 and let formula_13 denote the set of codes in formula_10 such that every element has exactly formula_15  nonzero entries. Denote by formula_16 the number of elements in formula_1. Then, we define formula_18 to be the largest size of a code with length formula_2 and minimum distance formula_4: Similarly, we define formula_22 to be the largest size of a code in formula_13: Theorem 1 (Johnson bound for formula_18): If formula_26,