MinT - Best Known (128, 128+57, s)-Nets in Base 3

Best Known (128, 128+57, s)-Nets in Base 3

(128, 128+57, 192)-Net over F₃ — Constructive and digital

Digital (128, 185, 192)-net over F₃, using

1 times m-reduction [i] based on digital (128, 186, 192)-net over F₃, using
- trace code for nets [i] based on digital (4, 62, 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]

(128, 128+57, 381)-Net over F₃ — Digital

Digital (128, 185, 381)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+57, 7687)-Net in Base 3 — Upper bound on s

There is no (128, 185, 7688)-net in base 3, because

1 times m-reduction [i] would yield (128, 184, 7688)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6177 727190 591341 039002 594589 388160 303628 839857 545513 796166 677497 434190 279188 468235 690817 > 3¹⁸⁴ [i]