Best Known (220−117, 220, s)-Nets in Base 2
(220−117, 220, 55)-Net over F2 — Constructive and digital
Digital (103, 220, 55)-net over F2, using
- t-expansion [i] based on digital (100, 220, 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
(220−117, 220, 65)-Net over F2 — Digital
Digital (103, 220, 65)-net over F2, using
- t-expansion [i] based on digital (95, 220, 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
(220−117, 220, 216)-Net over F2 — Upper bound on s (digital)
There is no digital (103, 220, 217)-net over F2, because
- 13 times m-reduction [i] would yield digital (103, 207, 217)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2207, 217, F2, 104) (dual of [217, 10, 105]-code), but
- residual code [i] would yield linear OA(2103, 112, F2, 52) (dual of [112, 9, 53]-code), but
- residual code [i] would yield linear OA(251, 59, F2, 26) (dual of [59, 8, 27]-code), but
- adding a parity check bit [i] would yield linear OA(252, 60, F2, 27) (dual of [60, 8, 28]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(252, 60, F2, 27) (dual of [60, 8, 28]-code), but
- residual code [i] would yield linear OA(251, 59, F2, 26) (dual of [59, 8, 27]-code), but
- residual code [i] would yield linear OA(2103, 112, F2, 52) (dual of [112, 9, 53]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2207, 217, F2, 104) (dual of [217, 10, 105]-code), but
(220−117, 220, 217)-Net in Base 2 — Upper bound on s
There is no (103, 220, 218)-net in base 2, because
- 15 times m-reduction [i] would yield (103, 205, 218)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2205, 218, S2, 102), but
- the linear programming bound shows that M ≥ 272331 004236 659599 976647 399713 241508 574642 785762 357467 384194 793472 / 3835 > 2205 [i]
- extracting embedded orthogonal array [i] would yield OA(2205, 218, S2, 102), but