MinT - Best Known (68, 145, s)-Nets in Base 3

Best Known (68, 145, s)-Nets in Base 3

(68, 145, 56)-Net over F₃ — Constructive and digital

Digital (68, 145, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 53, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 92, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(68, 145, 72)-Net over F₃ — Digital

Digital (68, 145, 72)-net over F₃, using

t-expansion [i] based on digital (67, 145, 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]

(68, 145, 446)-Net in Base 3 — Upper bound on s

There is no (68, 145, 447)-net in base 3, because

1 times m-reduction [i] would yield (68, 144, 447)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 527 299092 384160 699907 062254 845884 432553 837741 695298 065002 772449 062197 > 3¹⁴⁴ [i]