Best Known (210−116, 210, s)-Nets in Base 2
(210−116, 210, 53)-Net over F2 — Constructive and digital
Digital (94, 210, 53)-net over F2, using
- t-expansion [i] based on digital (90, 210, 53)-net over F2, using
- net from sequence [i] based on digital (90, 52)-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 4 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 (90, 52)-sequence over F2, using
(210−116, 210, 60)-Net over F2 — Digital
Digital (94, 210, 60)-net over F2, using
- t-expansion [i] based on digital (92, 210, 60)-net over F2, using
- net from sequence [i] based on digital (92, 59)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 92 and N(F) ≥ 60, using
- net from sequence [i] based on digital (92, 59)-sequence over F2, using
(210−116, 210, 198)-Net over F2 — Upper bound on s (digital)
There is no digital (94, 210, 199)-net over F2, because
- 20 times m-reduction [i] would yield digital (94, 190, 199)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2190, 199, F2, 96) (dual of [199, 9, 97]-code), but
- residual code [i] would yield linear OA(294, 102, F2, 48) (dual of [102, 8, 49]-code), but
- residual code [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- “vT4†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- residual code [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- residual code [i] would yield linear OA(294, 102, F2, 48) (dual of [102, 8, 49]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2190, 199, F2, 96) (dual of [199, 9, 97]-code), but
(210−116, 210, 200)-Net in Base 2 — Upper bound on s
There is no (94, 210, 201)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1922 030238 224627 435943 425514 864761 685297 438230 363667 930458 679184 > 2210 [i]