Best Known (31, 57, s)-Nets in Base 2
(31, 57, 25)-Net over F2 — Constructive and digital
Digital (31, 57, 25)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 11)-net over F2, using
- digital (11, 37, 14)-net over F2, using
- net from sequence [i] based on digital (11, 13)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 11 and N(F) ≥ 14, using
- net from sequence [i] based on digital (11, 13)-sequence over F2, using
(31, 57, 27)-Net over F2 — Digital
Digital (31, 57, 27)-net over F2, using
- net from sequence [i] based on digital (31, 26)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 31 and N(F) ≥ 27, using
(31, 57, 96)-Net over F2 — Upper bound on s (digital)
There is no digital (31, 57, 97)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(257, 97, F2, 26) (dual of [97, 40, 27]-code), but
- adding a parity check bit [i] would yield linear OA(258, 98, F2, 27) (dual of [98, 40, 28]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(258, 98, F2, 27) (dual of [98, 40, 28]-code), but
(31, 57, 97)-Net in Base 2 — Upper bound on s
There is no (31, 57, 98)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(257, 98, S2, 26), but
- the linear programming bound shows that M ≥ 796 660189 281003 951121 817121 767504 740352 / 5287 278143 181290 578125 > 257 [i]