MinT - Best Known (231−51, 231, s)-Nets in Base 3

Best Known (231−51, 231, s)-Nets in Base 3

(231−51, 231, 640)-Net over F₃ — Constructive and digital

Digital (180, 231, 640)-net over F₃, using

t-expansion [i] based on digital (179, 231, 640)-net over F₃, using
- 1 times m-reduction [i] based on digital (179, 232, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 58, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(231−51, 231, 1584)-Net over F₃ — Digital

Digital (180, 231, 1584)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−51, 231, 124745)-Net in Base 3 — Upper bound on s

There is no (180, 231, 124746)-net in base 3, because

1 times m-reduction [i] would yield (180, 230, 124746)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 689244 546721 711781 286203 137070 515899 227637 251806 379202 861927 079814 486960 045151 525727 149157 472120 584119 090981 > 3²³⁰ [i]