Best Known (128, 256, s)-Nets in Base 2
(128, 256, 57)-Net over F2 — Constructive and digital
Digital (128, 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
(128, 256, 81)-Net over F2 — Digital
Digital (128, 256, 81)-net over F2, using
- t-expansion [i] based on digital (126, 256, 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
(128, 256, 265)-Net over F2 — Upper bound on s (digital)
There is no digital (128, 256, 266)-net over F2, because
- extracting embedded orthogonal array [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
(128, 256, 279)-Net in Base 2 — Upper bound on s
There is no (128, 256, 280)-net in base 2, because
- 20 times m-reduction [i] would yield (128, 236, 280)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2236, 280, S2, 108), but
- the linear programming bound shows that M ≥ 41504 278412 335844 454047 491725 650645 432186 567323 902384 277507 042637 995046 295745 131031 035904 / 340479 332843 203125 > 2236 [i]
- extracting embedded orthogonal array [i] would yield OA(2236, 280, S2, 108), but