MinT - Best Known (219−56, 219, s)-Nets in Base 3

Best Known (219−56, 219, s)-Nets in Base 3

(219−56, 219, 288)-Net over F₃ — Constructive and digital

Digital (163, 219, 288)-net over F₃, using

9 times m-reduction [i] based on digital (163, 228, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 76, 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]

(219−56, 219, 839)-Net over F₃ — Digital

Digital (163, 219, 839)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−56, 219, 30433)-Net in Base 3 — Upper bound on s

There is no (163, 219, 30434)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 308 838122 077968 926284 519514 046926 292893 076367 028636 168163 138850 968732 766055 439366 607099 239485 323939 386905 > 3²¹⁹ [i]