MinT - Best Known (179−59, 179, s)-Nets in Base 3

Best Known (179−59, 179, s)-Nets in Base 3

(179−59, 179, 156)-Net over F₃ — Constructive and digital

Digital (120, 179, 156)-net over F₃, using

17 times m-reduction [i] based on digital (120, 196, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 98, 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]

(179−59, 179, 300)-Net over F₃ — Digital

Digital (120, 179, 300)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(179−59, 179, 4921)-Net in Base 3 — Upper bound on s

There is no (120, 179, 4922)-net in base 3, because

1 times m-reduction [i] would yield (120, 178, 4922)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 471884 486718 212061 778209 736574 544285 802384 235540 179722 023736 817756 546910 853512 133085 > 3¹⁷⁸ [i]