Best Known (82, 164, s)-Nets in Base 2
(82, 164, 51)-Net over F2 — Constructive and digital
Digital (82, 164, 51)-net over F2, using
- t-expansion [i] based on digital (80, 164, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(82, 164, 56)-Net over F2 — Digital
Digital (82, 164, 56)-net over F2, using
- t-expansion [i] based on digital (80, 164, 56)-net over F2, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
(82, 164, 177)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 164, 178)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2164, 178, F2, 82) (dual of [178, 14, 83]-code), but
- residual code [i] would yield linear OA(282, 95, F2, 41) (dual of [95, 13, 42]-code), but
- 1 times truncation [i] would yield linear OA(281, 94, F2, 40) (dual of [94, 13, 41]-code), but
- residual code [i] would yield linear OA(282, 95, F2, 41) (dual of [95, 13, 42]-code), but
(82, 164, 203)-Net in Base 2 — Upper bound on s
There is no (82, 164, 204)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 27 097609 928180 228136 086694 934662 947093 915640 634650 > 2164 [i]