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

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

(138, 138+61, 204)-Net over F₃ — Constructive and digital

Digital (138, 199, 204)-net over F₃, using

3¹ times duplication [i] based on digital (137, 198, 204)-net over F₃, using
- trace code for nets [i] based on digital (5, 66, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(138, 138+61, 412)-Net over F₃ — Digital

Digital (138, 199, 412)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+61, 8457)-Net in Base 3 — Upper bound on s

There is no (138, 199, 8458)-net in base 3, because

1 times m-reduction [i] would yield (138, 198, 8458)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29550 699232 913177 948045 432786 939424 994935 542408 534522 814623 626699 396912 181852 436978 967463 799005 > 3¹⁹⁸ [i]