MinT - Best Known (129, 129+75, s)-Nets in Base 3

Best Known (129, 129+75, s)-Nets in Base 3

(129, 129+75, 156)-Net over F₃ — Constructive and digital

Digital (129, 204, 156)-net over F₃, using

10 times m-reduction [i] based on digital (129, 214, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 107, 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]

(129, 129+75, 242)-Net over F₃ — Digital

Digital (129, 204, 242)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+75, 3001)-Net in Base 3 — Upper bound on s

There is no (129, 204, 3002)-net in base 3, because

1 times m-reduction [i] would yield (129, 203, 3002)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 191223 890402 205607 647144 586518 126337 210974 341396 365677 707149 919208 407552 952723 091390 435223 294477 > 3²⁰³ [i]