MinT - Best Known (128−49, 128, s)-Nets in Base 3

Best Known (128−49, 128, s)-Nets in Base 3

(128−49, 128, 128)-Net over F₃ — Constructive and digital

Digital (79, 128, 128)-net over F₃, using

4 times m-reduction [i] based on digital (79, 132, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 66, 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]

(128−49, 128, 145)-Net over F₃ — Digital

Digital (79, 128, 145)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−49, 128, 1617)-Net in Base 3 — Upper bound on s

There is no (79, 128, 1618)-net in base 3, because

1 times m-reduction [i] would yield (79, 127, 1618)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 952838 383878 905488 643658 407438 189302 757508 682903 469408 960497 > 3¹²⁷ [i]