MinT - Best Known (67, 67+130, s)-Nets in Base 3

Best Known (67, 67+130, s)-Nets in Base 3

(67, 67+130, 48)-Net over F₃ — Constructive and digital

Digital (67, 197, 48)-net over F₃, using

t-expansion [i] based on digital (45, 197, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(67, 67+130, 72)-Net over F₃ — Digital

Digital (67, 197, 72)-net over F₃, using

net from sequence [i] based on digital (67, 71)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(67, 67+130, 220)-Net over F₃ — Upper bound on s (digital)

There is no digital (67, 197, 221)-net over F₃, because

1 times m-reduction [i] would yield digital (67, 196, 221)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁶, 221, F₃, 129) (dual of [221, 25, 130]-code), but
  - residual code [i] would yield OA(3⁶⁷, 91, S₃, 43), but
    - the linear programming bound shows that MÂ â‰¥ 14877 993637 977476 431918 834321 910343 999390 452202 / 142 782236 368585 > 3⁶⁷ [i]

(67, 67+130, 289)-Net in Base 3 — Upper bound on s

There is no (67, 197, 290)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10340 501368 101962 677050 947309 932966 509529 684321 061789 099206 400924 528387 033722 804663 467492 438981 > 3¹⁹⁷ [i]