MinT - Best Known (68−43, 68, s)-Nets in Base 3

Best Known (68−43, 68, s)-Nets in Base 3

(68−43, 68, 32)-Net over F₃ — Constructive and digital

Digital (25, 68, 32)-net over F₃, using

t-expansion [i] based on digital (21, 68, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(68−43, 68, 36)-Net over F₃ — Digital

Digital (25, 68, 36)-net over F₃, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68−43, 68, 98)-Net in Base 3 — Upper bound on s

There is no (25, 68, 99)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁶⁸, 99, S₃, 43), but
- the linear programming bound shows that MÂ â‰¥ 5055 594369 602543 151259 965395 524976 895755 876197 115283 / 17 356329 984511 718750 > 3⁶⁸ [i]