MinT - Best Known (138, 138+105, s)-Nets in Base 3

Best Known (138, 138+105, s)-Nets in Base 3

(138, 138+105, 128)-Net over F₃ — Constructive and digital

Digital (138, 243, 128)-net over F₃, using

7 times m-reduction [i] based on digital (138, 250, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 125, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(138, 138+105, 179)-Net over F₃ — Digital

Digital (138, 243, 179)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+105, 1629)-Net in Base 3 — Upper bound on s

There is no (138, 243, 1630)-net in base 3, because

1 times m-reduction [i] would yield (138, 242, 1630)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 816300 442626 820984 569639 700626 978567 965139 066764 888477 047084 212980 039718 266322 138945 335369 653340 178459 282926 664169 > 3²⁴² [i]