Best Known (129, 129+128, s)-Nets in Base 2
(129, 129+128, 57)-Net over F2 — Constructive and digital
Digital (129, 257, 57)-net over F2, using
- t-expansion [i] based on digital (110, 257, 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
(129, 129+128, 81)-Net over F2 — Digital
Digital (129, 257, 81)-net over F2, using
- t-expansion [i] based on digital (126, 257, 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
(129, 129+128, 269)-Net over F2 — Upper bound on s (digital)
There is no digital (129, 257, 270)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2257, 270, F2, 128) (dual of [270, 13, 129]-code), but
- construction Y1 [i] would yield
- linear OA(2256, 266, F2, 128) (dual of [266, 10, 129]-code), but
- residual code [i] would yield linear OA(2128, 137, F2, 64) (dual of [137, 9, 65]-code), but
- residual code [i] would yield linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- residual code [i] would yield linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- residual code [i] would yield linear OA(2128, 137, F2, 64) (dual of [137, 9, 65]-code), but
- OA(213, 270, S2, 4), but
- discarding factors would yield OA(213, 128, S2, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 8257 > 213 [i]
- discarding factors would yield OA(213, 128, S2, 4), but
- linear OA(2256, 266, F2, 128) (dual of [266, 10, 129]-code), but
- construction Y1 [i] would yield
(129, 129+128, 284)-Net in Base 2 — Upper bound on s
There is no (129, 257, 285)-net in base 2, because
- 20 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