MinT - Best Known (228−83, 228, s)-Nets in Base 3

Best Known (228−83, 228, s)-Nets in Base 3

(228−83, 228, 156)-Net over F₃ — Constructive and digital

Digital (145, 228, 156)-net over F₃, using

18 times m-reduction [i] based on digital (145, 246, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 123, 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]

(228−83, 228, 275)-Net over F₃ — Digital

Digital (145, 228, 275)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−83, 228, 3495)-Net in Base 3 — Upper bound on s

There is no (145, 228, 3496)-net in base 3, because

1 times m-reduction [i] would yield (145, 227, 3496)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 034005 408132 536847 775262 659464 096258 372479 701274 467942 861573 583735 739665 952071 176496 425721 007341 882674 640593 > 3²²⁷ [i]