MinT - Best Known (163, 163+45, s)-Nets in Base 3

Best Known (163, 163+45, s)-Nets in Base 3

(163, 163+45, 688)-Net over F₃ — Constructive and digital

Digital (163, 208, 688)-net over F₃, using

trace code for nets [i] based on digital (7, 52, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(163, 163+45, 1575)-Net over F₃ — Digital

Digital (163, 208, 1575)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+45, 139647)-Net in Base 3 — Upper bound on s

There is no (163, 208, 139648)-net in base 3, because

1 times m-reduction [i] would yield (163, 207, 139648)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 580 935535 540405 664647 370360 289459 849513 077572 622943 694778 025915 898617 526115 323976 086624 122695 158017 > 3²⁰⁷ [i]