Best Known (85, 85+96, s)-Nets in Base 2
(85, 85+96, 52)-Net over F2 — Constructive and digital
Digital (85, 181, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-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 3 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]
(85, 85+96, 57)-Net over F2 — Digital
Digital (85, 181, 57)-net over F2, using
- t-expansion [i] based on digital (83, 181, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(85, 85+96, 181)-Net over F2 — Upper bound on s (digital)
There is no digital (85, 181, 182)-net over F2, because
- 8 times m-reduction [i] would yield digital (85, 173, 182)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2173, 182, F2, 88) (dual of [182, 9, 89]-code), but
- residual code [i] would yield linear OA(285, 93, F2, 44) (dual of [93, 8, 45]-code), but
- residual code [i] would yield linear OA(241, 48, F2, 22) (dual of [48, 7, 23]-code), but
- residual code [i] would yield linear OA(219, 25, F2, 11) (dual of [25, 6, 12]-code), but
- 1 times truncation [i] would yield linear OA(218, 24, F2, 10) (dual of [24, 6, 11]-code), but
- residual code [i] would yield linear OA(28, 13, F2, 5) (dual of [13, 5, 6]-code), but
- 1 times truncation [i] would yield linear OA(27, 12, F2, 4) (dual of [12, 5, 5]-code), but
- residual code [i] would yield linear OA(28, 13, F2, 5) (dual of [13, 5, 6]-code), but
- 1 times truncation [i] would yield linear OA(218, 24, F2, 10) (dual of [24, 6, 11]-code), but
- residual code [i] would yield linear OA(219, 25, F2, 11) (dual of [25, 6, 12]-code), but
- residual code [i] would yield linear OA(241, 48, F2, 22) (dual of [48, 7, 23]-code), but
- residual code [i] would yield linear OA(285, 93, F2, 44) (dual of [93, 8, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2173, 182, F2, 88) (dual of [182, 9, 89]-code), but
(85, 85+96, 182)-Net in Base 2 — Upper bound on s
There is no (85, 181, 183)-net in base 2, because
- 4 times m-reduction [i] would yield (85, 177, 183)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2177, 183, S2, 92), but
- the (dual) Plotkin bound shows that M ≥ 6 129982 163463 555433 433388 108601 236734 474956 488734 408704 / 31 > 2177 [i]
- extracting embedded orthogonal array [i] would yield OA(2177, 183, S2, 92), but