Best Known (72, 72+78, s)-Nets in Base 2
(72, 72+78, 49)-Net over F2 — Constructive and digital
Digital (72, 150, 49)-net over F2, using
- t-expansion [i] based on digital (70, 150, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
(72, 72+78, 154)-Net over F2 — Upper bound on s (digital)
There is no digital (72, 150, 155)-net over F2, because
- 6 times m-reduction [i] would yield digital (72, 144, 155)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2144, 155, F2, 72) (dual of [155, 11, 73]-code), but
- residual code [i] would yield linear OA(272, 82, F2, 36) (dual of [82, 10, 37]-code), but
- adding a parity check bit [i] would yield linear OA(273, 83, F2, 37) (dual of [83, 10, 38]-code), but
- residual code [i] would yield linear OA(272, 82, F2, 36) (dual of [82, 10, 37]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2144, 155, F2, 72) (dual of [155, 11, 73]-code), but
(72, 72+78, 155)-Net in Base 2 — Upper bound on s
There is no (72, 150, 156)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2150, 156, S2, 78), but
- adding a parity check bit [i] would yield OA(2151, 157, S2, 79), but
- the (dual) Plotkin bound shows that M ≥ 17126 972312 471518 572699 431633 393941 636592 959488 / 5 > 2151 [i]
- adding a parity check bit [i] would yield OA(2151, 157, S2, 79), but