Best Known (81, 81+87, s)-Nets in Base 2
(81, 81+87, 51)-Net over F2 — Constructive and digital
Digital (81, 168, 51)-net over F2, using
- t-expansion [i] based on digital (80, 168, 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+87, 56)-Net over F2 — Digital
Digital (81, 168, 56)-net over F2, using
- t-expansion [i] based on digital (80, 168, 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+87, 172)-Net over F2 — Upper bound on s (digital)
There is no digital (81, 168, 173)-net over F2, because
- 3 times m-reduction [i] would yield digital (81, 165, 173)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2165, 173, F2, 84) (dual of [173, 8, 85]-code), but
- residual code [i] would yield linear OA(281, 88, F2, 42) (dual of [88, 7, 43]-code), but
- “Hel†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(281, 88, F2, 42) (dual of [88, 7, 43]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2165, 173, F2, 84) (dual of [173, 8, 85]-code), but
(81, 81+87, 191)-Net in Base 2 — Upper bound on s
There is no (81, 168, 192)-net in base 2, because
- 1 times m-reduction [i] would yield (81, 167, 192)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 192 965522 211198 158714 497181 340705 946559 859730 020255 > 2167 [i]