Best Known (165−86, 165, s)-Nets in Base 2
(165−86, 165, 50)-Net over F2 — Constructive and digital
Digital (79, 165, 50)-net over F2, using
- t-expansion [i] based on digital (75, 165, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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 (75, 49)-sequence over F2, using
(165−86, 165, 52)-Net over F2 — Digital
Digital (79, 165, 52)-net over F2, using
- t-expansion [i] based on digital (77, 165, 52)-net over F2, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 77 and N(F) ≥ 52, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
(165−86, 165, 167)-Net over F2 — Upper bound on s (digital)
There is no digital (79, 165, 168)-net over F2, because
- 6 times m-reduction [i] would yield digital (79, 159, 168)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2159, 168, F2, 80) (dual of [168, 9, 81]-code), but
- residual code [i] would yield linear OA(279, 87, F2, 40) (dual of [87, 8, 41]-code), but
- adding a parity check bit [i] would yield linear OA(280, 88, F2, 41) (dual of [88, 8, 42]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(280, 88, F2, 41) (dual of [88, 8, 42]-code), but
- residual code [i] would yield linear OA(279, 87, F2, 40) (dual of [87, 8, 41]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2159, 168, F2, 80) (dual of [168, 9, 81]-code), but
(165−86, 165, 170)-Net in Base 2 — Upper bound on s
There is no (79, 165, 171)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2165, 171, S2, 86), but
- the (dual) Plotkin bound shows that M ≥ 1496 577676 626844 588240 573268 701473 812127 674924 007424 / 29 > 2165 [i]