Best Known (85, 85+86, s)-Nets in Base 2
(85, 85+86, 52)-Net over F2 — Constructive and digital
Digital (85, 171, 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+86, 57)-Net over F2 — Digital
Digital (85, 171, 57)-net over F2, using
- t-expansion [i] based on digital (83, 171, 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+86, 182)-Net over F2 — Upper bound on s (digital)
There is no digital (85, 171, 183)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2171, 183, F2, 86) (dual of [183, 12, 87]-code), but
- residual code [i] would yield linear OA(285, 96, F2, 43) (dual of [96, 11, 44]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(285, 96, F2, 43) (dual of [96, 11, 44]-code), but
(85, 85+86, 208)-Net in Base 2 — Upper bound on s
There is no (85, 171, 209)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3497 037589 097971 479633 737055 886313 806109 084230 366560 > 2171 [i]