Best Known (254−125, 254, s)-Nets in Base 2
(254−125, 254, 57)-Net over F2 — Constructive and digital
Digital (129, 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
(254−125, 254, 81)-Net over F2 — Digital
Digital (129, 254, 81)-net over F2, using
- t-expansion [i] based on digital (126, 254, 81)-net over F2, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
(254−125, 254, 276)-Net over F2 — Upper bound on s (digital)
There is no digital (129, 254, 277)-net over F2, because
- 1 times m-reduction [i] would yield digital (129, 253, 277)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2253, 277, F2, 124) (dual of [277, 24, 125]-code), but
- residual code [i] would yield linear OA(2129, 152, F2, 62) (dual of [152, 23, 63]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- adding a parity check bit [i] would yield linear OA(2131, 154, F2, 63) (dual of [154, 23, 64]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- residual code [i] would yield linear OA(2129, 152, F2, 62) (dual of [152, 23, 63]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2253, 277, F2, 124) (dual of [277, 24, 125]-code), but
(254−125, 254, 284)-Net in Base 2 — Upper bound on s
There is no (129, 254, 285)-net in base 2, because
- 17 times m-reduction [i] would yield (129, 237, 285)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2237, 285, S2, 108), but
- the linear programming bound shows that M ≥ 56343 399996 195265 169227 114420 218575 369487 305782 122681 969429 903696 483971 902670 896111 812608 / 204485 216198 046875 > 2237 [i]
- extracting embedded orthogonal array [i] would yield OA(2237, 285, S2, 108), but