MinT - Best Known (165−51, 165, s)-Nets in Base 3

Best Known (165−51, 165, s)-Nets in Base 3

(165−51, 165, 192)-Net over F₃ — Constructive and digital

Digital (114, 165, 192)-net over F₃, using

trace code for nets [i] based on digital (4, 55, 64)-net over F₂₇, using
- net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
    - a function field by SÃ©mirat [i]

(165−51, 165, 342)-Net over F₃ — Digital

Digital (114, 165, 342)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−51, 165, 6838)-Net in Base 3 — Upper bound on s

There is no (114, 165, 6839)-net in base 3, because

1 times m-reduction [i] would yield (114, 164, 6839)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 772913 013232 815423 236883 268392 305524 310665 760387 008158 446750 921022 348223 867199 > 3¹⁶⁴ [i]