MinT - Best Known (138−61, 138, s)-Nets in Base 3

Best Known (138−61, 138, s)-Nets in Base 3

(138−61, 138, 80)-Net over F₃ — Constructive and digital

Digital (77, 138, 80)-net over F₃, using

trace code for nets [i] based on digital (8, 69, 40)-net over F₉, using
- net from sequence [i] based on digital (8, 39)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
    - a function field by SÃ©mirat [i]

(138−61, 138, 103)-Net over F₃ — Digital

Digital (77, 138, 103)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−61, 138, 879)-Net in Base 3 — Upper bound on s

There is no (77, 138, 880)-net in base 3, because

1 times m-reduction [i] would yield (77, 137, 880)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 232108 420152 092708 218671 558034 122727 525677 692372 374738 303516 429729 > 3¹³⁷ [i]