MinT - Best Known (163−90, 163, s)-Nets in Base 3

Best Known (163−90, 163, s)-Nets in Base 3

(163−90, 163, 52)-Net over F₃ — Constructive and digital

Digital (73, 163, 52)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 58, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (15, 105, 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]

(163−90, 163, 84)-Net over F₃ — Digital

Digital (73, 163, 84)-net over F₃, using

t-expansion [i] based on digital (71, 163, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(163−90, 163, 428)-Net in Base 3 — Upper bound on s

There is no (73, 163, 429)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 615322 833456 904075 507137 714656 809385 292277 828692 530811 450833 861628 771985 028427 > 3¹⁶³ [i]