Best Known (81, 81+80, s)-Nets in Base 2
(81, 81+80, 51)-Net over F2 — Constructive and digital
Digital (81, 161, 51)-net over F2, using
- t-expansion [i] based on digital (80, 161, 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
(81, 81+80, 56)-Net over F2 — Digital
Digital (81, 161, 56)-net over F2, using
- t-expansion [i] based on digital (80, 161, 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
(81, 81+80, 174)-Net over F2 — Upper bound on s (digital)
There is no digital (81, 161, 175)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2161, 175, F2, 80) (dual of [175, 14, 81]-code), but
- residual code [i] would yield linear OA(281, 94, F2, 40) (dual of [94, 13, 41]-code), but
(81, 81+80, 202)-Net in Base 2 — Upper bound on s
There is no (81, 161, 203)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 012513 351737 896627 449195 291538 903480 394038 336050 > 2161 [i]