MinT - Best Known (159, 159+45, s)-Nets in Base 3

Best Known (159, 159+45, s)-Nets in Base 3

(159, 159+45, 640)-Net over F₃ — Constructive and digital

Digital (159, 204, 640)-net over F₃, using

t-expansion [i] based on digital (158, 204, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 51, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(159, 159+45, 1428)-Net over F₃ — Digital

Digital (159, 204, 1428)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+45, 114358)-Net in Base 3 — Upper bound on s

There is no (159, 204, 114359)-net in base 3, because

1 times m-reduction [i] would yield (159, 203, 114359)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 171945 255928 487064 051052 457538 146170 983986 510230 696778 180432 675176 623044 801659 233776 072091 848069 > 3²⁰³ [i]