MinT - Best Known (128, 128+67, s)-Nets in Base 3

Best Known (128, 128+67, s)-Nets in Base 3

(128, 128+67, 156)-Net over F₃ — Constructive and digital

Digital (128, 195, 156)-net over F₃, using

17 times m-reduction [i] based on digital (128, 212, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 106, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(128, 128+67, 284)-Net over F₃ — Digital

Digital (128, 195, 284)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+67, 4167)-Net in Base 3 — Upper bound on s

There is no (128, 195, 4168)-net in base 3, because

1 times m-reduction [i] would yield (128, 194, 4168)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 366 336243 244591 010500 513401 880762 568373 600720 953773 662764 528360 879598 605724 786034 693589 588625 > 3¹⁹⁴ [i]