MinT - Best Known (223−105, 223, s)-Nets in Base 3

Best Known (223−105, 223, s)-Nets in Base 3

(223−105, 223, 80)-Net over F₃ — Constructive and digital

Digital (118, 223, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 73, 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 (45, 150, 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]

(223−105, 223, 130)-Net over F₃ — Digital

Digital (118, 223, 130)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−105, 223, 1050)-Net in Base 3 — Upper bound on s

There is no (118, 223, 1051)-net in base 3, because

1 times m-reduction [i] would yield (118, 222, 1051)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8521 902881 961924 751762 141935 479119 364042 644802 062746 917274 539249 308417 439582 109723 775390 391842 297605 527673 > 3²²² [i]