MinT - Best Known (128, 128+59, s)-Nets in Base 3

Best Known (128, 128+59, s)-Nets in Base 3

(128, 128+59, 162)-Net over F₃ — Constructive and digital

Digital (128, 187, 162)-net over F₃, using

5 times m-reduction [i] based on digital (128, 192, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 96, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(128, 128+59, 357)-Net over F₃ — Digital

Digital (128, 187, 357)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+59, 6674)-Net in Base 3 — Upper bound on s

There is no (128, 187, 6675)-net in base 3, because

1 times m-reduction [i] would yield (128, 186, 6675)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55716 384494 578649 533320 361358 875777 822611 459665 013678 696172 200776 400008 578670 160678 008495 > 3¹⁸⁶ [i]