Best Known (65, 65+69, s)-Nets in Base 2
(65, 65+69, 43)-Net over F2 — Constructive and digital
Digital (65, 134, 43)-net over F2, using
- t-expansion [i] based on digital (59, 134, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(65, 65+69, 48)-Net over F2 — Digital
Digital (65, 134, 48)-net over F2, using
- net from sequence [i] based on digital (65, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 65 and N(F) ≥ 48, using
(65, 65+69, 139)-Net over F2 — Upper bound on s (digital)
There is no digital (65, 134, 140)-net over F2, because
- 3 times m-reduction [i] would yield digital (65, 131, 140)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2131, 140, F2, 66) (dual of [140, 9, 67]-code), but
- residual code [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2131, 140, F2, 66) (dual of [140, 9, 67]-code), but
(65, 65+69, 141)-Net in Base 2 — Upper bound on s
There is no (65, 134, 142)-net in base 2, because
- 3 times m-reduction [i] would yield (65, 131, 142)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2131, 142, S2, 66), but
- the linear programming bound shows that M ≥ 10 529697 562001 519813 410663 852368 635535 294464 / 3485 > 2131 [i]
- extracting embedded orthogonal array [i] would yield OA(2131, 142, S2, 66), but