Best Known (27, 62, s)-Nets in Base 2
(27, 62, 21)-Net over F2 — Constructive and digital
Digital (27, 62, 21)-net over F2, using
- t-expansion [i] based on digital (21, 62, 21)-net over F2, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 21 and N(F) ≥ 21, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
(27, 62, 24)-Net over F2 — Digital
Digital (27, 62, 24)-net over F2, using
- t-expansion [i] based on digital (25, 62, 24)-net over F2, using
- net from sequence [i] based on digital (25, 23)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 25 and N(F) ≥ 24, using
- net from sequence [i] based on digital (25, 23)-sequence over F2, using
(27, 62, 62)-Net over F2 — Upper bound on s (digital)
There is no digital (27, 62, 63)-net over F2, because
- 7 times m-reduction [i] would yield digital (27, 55, 63)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(255, 63, F2, 28) (dual of [63, 8, 29]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- extracting embedded orthogonal array [i] would yield linear OA(255, 63, F2, 28) (dual of [63, 8, 29]-code), but
(27, 62, 63)-Net in Base 2 — Upper bound on s
There is no (27, 62, 64)-net in base 2, because
- 3 times m-reduction [i] would yield (27, 59, 64)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(259, 64, S2, 32), but
- adding a parity check bit [i] would yield OA(260, 65, S2, 33), but
- the (dual) Plotkin bound shows that M ≥ 27 670116 110564 327424 / 17 > 260 [i]
- adding a parity check bit [i] would yield OA(260, 65, S2, 33), but
- extracting embedded orthogonal array [i] would yield OA(259, 64, S2, 32), but