MinT - Best Known (83, 83+73, s)-Nets in Base 3

Best Known (83, 83+73, s)-Nets in Base 3

(83, 83+73, 68)-Net over F₃ — Constructive and digital

Digital (83, 156, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 57, 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]
2. digital (26, 99, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(83, 83+73, 97)-Net over F₃ — Digital

Digital (83, 156, 97)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+73, 774)-Net in Base 3 — Upper bound on s

There is no (83, 156, 775)-net in base 3, because

1 times m-reduction [i] would yield (83, 155, 775)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 93 198961 970483 910185 836484 547950 928145 861139 067565 493700 102321 229678 770233 > 3¹⁵⁵ [i]