MinT - Best Known (239−85, 239, s)-Nets in Base 3

Best Known (239−85, 239, s)-Nets in Base 3

(239−85, 239, 162)-Net over F₃ — Constructive and digital

Digital (154, 239, 162)-net over F₃, using

5 times m-reduction [i] based on digital (154, 244, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 122, 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]

(239−85, 239, 306)-Net over F₃ — Digital

Digital (154, 239, 306)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−85, 239, 4131)-Net in Base 3 — Upper bound on s

There is no (154, 239, 4132)-net in base 3, because

1 times m-reduction [i] would yield (154, 238, 4132)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 359625 289293 728104 597186 175582 908619 861459 362957 154056 101121 969046 146705 605578 054453 654790 237734 738743 352936 606073 > 3²³⁸ [i]