Best Known (212−110, 212, s)-Nets in Base 2
(212−110, 212, 55)-Net over F2 — Constructive and digital
Digital (102, 212, 55)-net over F2, using
- t-expansion [i] based on digital (100, 212, 55)-net over F2, using
- net from sequence [i] based on digital (100, 54)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 6 places with degree 6 [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 (100, 54)-sequence over F2, using
(212−110, 212, 65)-Net over F2 — Digital
Digital (102, 212, 65)-net over F2, using
- t-expansion [i] based on digital (95, 212, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(212−110, 212, 214)-Net over F2 — Upper bound on s (digital)
There is no digital (102, 212, 215)-net over F2, because
- 6 times m-reduction [i] would yield digital (102, 206, 215)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2206, 215, F2, 104) (dual of [215, 9, 105]-code), but
- residual code [i] would yield linear OA(2102, 110, F2, 52) (dual of [110, 8, 53]-code), but
- residual code [i] would yield linear OA(250, 57, F2, 26) (dual of [57, 7, 27]-code), but
- adding a parity check bit [i] would yield linear OA(251, 58, F2, 27) (dual of [58, 7, 28]-code), but
- “vT3†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(251, 58, F2, 27) (dual of [58, 7, 28]-code), but
- residual code [i] would yield linear OA(250, 57, F2, 26) (dual of [57, 7, 27]-code), but
- residual code [i] would yield linear OA(2102, 110, F2, 52) (dual of [110, 8, 53]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2206, 215, F2, 104) (dual of [215, 9, 105]-code), but
(212−110, 212, 215)-Net in Base 2 — Upper bound on s
There is no (102, 212, 216)-net in base 2, because
- 10 times m-reduction [i] would yield (102, 202, 216)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2202, 216, S2, 100), but
- the linear programming bound shows that M ≥ 233661 647139 611257 986005 623923 143771 707548 492919 968340 595502 481408 / 29087 > 2202 [i]
- extracting embedded orthogonal array [i] would yield OA(2202, 216, S2, 100), but