MinT - Best Known (45−21, 45, s)-Nets in Base 3

Best Known (45−21, 45, s)-Nets in Base 3

(45−21, 45, 34)-Net over F₃ — Constructive and digital

Digital (24, 45, 34)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 15)-net over F₃, using
  - a shift-net [i]
2. digital (9, 30, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]

(45−21, 45, 39)-Net over F₃ — Digital

Digital (24, 45, 39)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(45−21, 45, 275)-Net in Base 3 — Upper bound on s

There is no (24, 45, 276)-net in base 3, because

1 times m-reduction [i] would yield (24, 44, 276)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1010 674657 351427 105049 > 3⁴⁴ [i]