Design Theory, 2nd Edition

 

10.4 The (3v - 2u)‐Construction

In this section we present a recursive construction that starts with an SQS (v) (V = { 1 , 2 , . . . , v } , B) that contains an S Q S ( u ) ( V ′ = { v - u + 1 , . . . , v } , B ′ ) and produces an SQS(3v - 2u)(V B where V ″ = { ∞ 1 , . . . , ∞ U } ∪ ( { 1 , 2 , … , v - u } × {1, 2, 3}). This result was originally obtained by Hartman, but here we present a simplified proof due to Lenz, who made use of the Stern‐Lenz Lemma.

It will simplify notation to define g = v - u for the rest of this chapter. Notice that if 1 ≤ a ≤ g then a naturally corresponds to 3 symbols in V namely ( a , 1 ) , ( a , 2 ) and ( a , 3 ) ; and if g + 1 ≤ a ≤ v then symbol a in the SQS(v) ...