Best Known (20, 20+27, s)-Nets in Base 2
(20, 20+27, 20)-Net over F2 — Constructive and digital
Digital (20, 47, 20)-net over F2, using
- t-expansion [i] based on digital (19, 47, 20)-net over F2, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 19 and N(F) ≥ 20, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
(20, 20+27, 47)-Net over F2 — Upper bound on s (digital)
There is no digital (20, 47, 48)-net over F2, because
- 3 times m-reduction [i] would yield digital (20, 44, 48)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(244, 48, F2, 24) (dual of [48, 4, 25]-code), but
(20, 20+27, 48)-Net in Base 2 — Upper bound on s
There is no (20, 47, 49)-net in base 2, because
- 3 times m-reduction [i] would yield (20, 44, 49)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(244, 49, S2, 24), but
- adding a parity check bit [i] would yield OA(245, 50, S2, 25), but
- the (dual) Plotkin bound shows that M ≥ 562 949953 421312 / 13 > 245 [i]
- adding a parity check bit [i] would yield OA(245, 50, S2, 25), but
- extracting embedded orthogonal array [i] would yield OA(244, 49, S2, 24), but