MinT - Best Known (172, 172+61, s)-Nets in Base 3

Best Known (172, 172+61, s)-Nets in Base 3

(172, 172+61, 288)-Net over F₃ — Constructive and digital

Digital (172, 233, 288)-net over F₃, using

t-expansion [i] based on digital (171, 233, 288)-net over F₃, using
- 7 times m-reduction [i] based on digital (171, 240, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 80, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]

(172, 172+61, 809)-Net over F₃ — Digital

Digital (172, 233, 809)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+61, 29448)-Net in Base 3 — Upper bound on s

There is no (172, 233, 29449)-net in base 3, because

1 times m-reduction [i] would yield (172, 232, 29449)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 492 488134 046510 081419 024904 909804 425016 253079 965343 578511 864132 662427 032361 180142 568993 135541 891606 075294 205577 > 3²³² [i]