MinT - Best Known (247−115, 247, s)-Nets in Base 3

Best Known (247−115, 247, s)-Nets in Base 3

(247−115, 247, 85)-Net over F₃ — Constructive and digital

Digital (132, 247, 85)-net over F₃, using

t-expansion [i] based on digital (131, 247, 85)-net over F₃, using
- 2 times m-reduction [i] based on digital (131, 249, 85)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (27, 86, 37)-net over F₃, using
      - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
    2. digital (45, 163, 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]

(247−115, 247, 147)-Net over F₃ — Digital

Digital (132, 247, 147)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−115, 247, 1209)-Net in Base 3 — Upper bound on s

There is no (132, 247, 1210)-net in base 3, because

1 times m-reduction [i] would yield (132, 246, 1210)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2395 458095 519126 016607 174522 288867 543518 118585 450828 487267 675179 432708 393369 659624 823793 327248 556605 500306 286084 267205 > 3²⁴⁶ [i]