ICT12 3. Channel coding

15If C = {x 1 , … , x M }, let D i = B(x i , t), where t = ⌊(d − 1)/2⌋, with d theminimum distance of C. So t is the greatest integer such that 2t ≤ d − 1,or 2t < d.d 3 4 5 6 7t 1 1 2 2 3Since hdx j , x k ≥ d for j ≠ k, it follows that C ∩ D i = {x i }.It is also true that D j ∩ D k = ∅ if j ≠ k. Indeed, if y ∈ D j ∩ D k , thenhdx j , x k ≤ hdx j , y + hd(y, x k ) ≤ t + t < d.But by definition of d this can only occur if x j = x k .