Best Known (256−135, 256, s)-Nets in Base 2
(256−135, 256, 57)-Net over F2 — Constructive and digital
Digital (121, 256, 57)-net over F2, using
- t-expansion [i] based on digital (110, 256, 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
(256−135, 256, 80)-Net over F2 — Digital
Digital (121, 256, 80)-net over F2, using
- net from sequence [i] based on digital (121, 79)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 121 and N(F) ≥ 80, using
(256−135, 256, 254)-Net over F2 — Upper bound on s (digital)
There is no digital (121, 256, 255)-net over F2, because
- 11 times m-reduction [i] would yield digital (121, 245, 255)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2245, 255, F2, 124) (dual of [255, 10, 125]-code), but
- residual code [i] would yield linear OA(2121, 130, F2, 62) (dual of [130, 9, 63]-code), but
- residual code [i] would yield linear OA(259, 67, F2, 31) (dual of [67, 8, 32]-code), but
- “DHM†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(259, 67, F2, 31) (dual of [67, 8, 32]-code), but
- residual code [i] would yield linear OA(2121, 130, F2, 62) (dual of [130, 9, 63]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2245, 255, F2, 124) (dual of [255, 10, 125]-code), but
(256−135, 256, 255)-Net in Base 2 — Upper bound on s
There is no (121, 256, 256)-net in base 2, because
- 7 times m-reduction [i] would yield (121, 249, 256)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2249, 256, S2, 128), but
- adding a parity check bit [i] would yield OA(2250, 257, S2, 129), but
- the (dual) Plotkin bound shows that M ≥ 173688 133855 974293 135356 477513 031861 779904 976998 460846 059186 376011 869694 459904 / 65 > 2250 [i]
- adding a parity check bit [i] would yield OA(2250, 257, S2, 129), but
- extracting embedded orthogonal array [i] would yield OA(2249, 256, S2, 128), but