MinT - Best Known (245−91, 245, s)-Nets in Base 3

Best Known (245−91, 245, s)-Nets in Base 3

(245−91, 245, 156)-Net over F₃ — Constructive and digital

Digital (154, 245, 156)-net over F₃, using

t-expansion [i] based on digital (147, 245, 156)-net over F₃, using
- 5 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
  - trace code for nets [i] based on digital (22, 125, 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]

(245−91, 245, 276)-Net over F₃ — Digital

Digital (154, 245, 276)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−91, 245, 3361)-Net in Base 3 — Upper bound on s

There is no (154, 245, 3362)-net in base 3, because

1 times m-reduction [i] would yield (154, 244, 3362)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 263 653399 329004 919958 082964 174363 244147 232977 770226 699770 053128 369315 707761 662594 642039 053080 693652 728657 879750 638669 > 3²⁴⁴ [i]