Best Known (127−69, 127, s)-Nets in Base 2
(127−69, 127, 42)-Net over F2 — Constructive and digital
Digital (58, 127, 42)-net over F2, using
- t-expansion [i] based on digital (54, 127, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
(127−69, 127, 126)-Net over F2 — Upper bound on s (digital)
There is no digital (58, 127, 127)-net over F2, because
- 9 times m-reduction [i] would yield digital (58, 118, 127)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2118, 127, F2, 60) (dual of [127, 9, 61]-code), but
- residual code [i] would yield linear OA(258, 66, F2, 30) (dual of [66, 8, 31]-code), but
- adding a parity check bit [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]
- adding a parity check bit [i] would yield linear OA(259, 67, F2, 31) (dual of [67, 8, 32]-code), but
- residual code [i] would yield linear OA(258, 66, F2, 30) (dual of [66, 8, 31]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2118, 127, F2, 60) (dual of [127, 9, 61]-code), but
(127−69, 127, 127)-Net in Base 2 — Upper bound on s
There is no (58, 127, 128)-net in base 2, because
- 11 times m-reduction [i] would yield (58, 116, 128)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2116, 128, S2, 58), but
- the linear programming bound shows that M ≥ 156 848903 502620 073002 649233 113080 659968 / 1575 > 2116 [i]
- extracting embedded orthogonal array [i] would yield OA(2116, 128, S2, 58), but