Best Known (254−135, 254, s)-Nets in Base 2
(254−135, 254, 57)-Net over F2 — Constructive and digital
Digital (119, 254, 57)-net over F2, using
- t-expansion [i] based on digital (110, 254, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-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 8 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 (110, 56)-sequence over F2, using
(254−135, 254, 73)-Net over F2 — Digital
Digital (119, 254, 73)-net over F2, using
- t-expansion [i] based on digital (114, 254, 73)-net over F2, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 114 and N(F) ≥ 73, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
(254−135, 254, 249)-Net over F2 — Upper bound on s (digital)
There is no digital (119, 254, 250)-net over F2, because
- 15 times m-reduction [i] would yield digital (119, 239, 250)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2239, 250, F2, 120) (dual of [250, 11, 121]-code), but
- residual code [i] would yield linear OA(2119, 129, F2, 60) (dual of [129, 10, 61]-code), but
- residual code [i] would yield linear OA(259, 68, F2, 30) (dual of [68, 9, 31]-code), but
- adding a parity check bit [i] would yield linear OA(260, 69, F2, 31) (dual of [69, 9, 32]-code), but
- “BGV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(260, 69, F2, 31) (dual of [69, 9, 32]-code), but
- residual code [i] would yield linear OA(259, 68, F2, 30) (dual of [68, 9, 31]-code), but
- residual code [i] would yield linear OA(2119, 129, F2, 60) (dual of [129, 10, 61]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2239, 250, F2, 120) (dual of [250, 11, 121]-code), but
(254−135, 254, 251)-Net in Base 2 — Upper bound on s
There is no (119, 254, 252)-net in base 2, because
- 9 times m-reduction [i] would yield (119, 245, 252)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2245, 252, S2, 126), but
- the (dual) Plotkin bound shows that M ≥ 7237 005577 332262 213973 186563 042994 240829 374041 602535 252466 099000 494570 602496 / 127 > 2245 [i]
- extracting embedded orthogonal array [i] would yield OA(2245, 252, S2, 126), but