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. 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. Let C {displaystyle C} be a q-ary code of length n {displaystyle n} , i.e. a subset of F q n {displaystyle mathbb {F} _{q}^{n}} . Let d {displaystyle d} be the minimum distance of C {displaystyle C} , i.e. where d ( x , y ) {displaystyle d(x,y)} is the Hamming distance between x {displaystyle x} and y {displaystyle y} . Let C q ( n , d ) {displaystyle C_{q}(n,d)} be the set of all q-ary codes with length n {displaystyle n} and minimum distance d {displaystyle d} and let C q ( n , d , w ) {displaystyle C_{q}(n,d,w)} denote the set of codes in C q ( n , d ) {displaystyle C_{q}(n,d)} such that every element has exactly w {displaystyle w} nonzero entries. Denote by | C | {displaystyle |C|} the number of elements in C {displaystyle C} . Then, we define A q ( n , d ) {displaystyle A_{q}(n,d)} to be the largest size of a code with length n {displaystyle n} and minimum distance d {displaystyle d} : Similarly, we define A q ( n , d , w ) {displaystyle A_{q}(n,d,w)} to be the largest size of a code in C q ( n , d , w ) {displaystyle C_{q}(n,d,w)} : Theorem 1 (Johnson bound for A q ( n , d ) {displaystyle A_{q}(n,d)} ): If d = 2 t + 1 {displaystyle d=2t+1} , If d = 2 t {displaystyle d=2t} ,