MinT - Best Known (113, 140, s)-Nets in Base 4

Best Known (113, 140, s)-Nets in Base 4

(113, 140, 1104)-Net over F₄ — Constructive and digital

Digital (113, 140, 1104)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (19, 32, 76)-net over F₄, using
  - trace code for nets [i] based on digital (3, 16, 38)-net over F₁₆, using
    - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (81, 108, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 27, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(113, 140, 6151)-Net over F₄ — Digital

Digital (113, 140, 6151)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(113, 140, 5172431)-Net in Base 4 — Upper bound on s

There is no (113, 140, 5172432)-net in base 4, because

1 times m-reduction [i] would yield (113, 139, 5172432)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 485668 059256 102085 687915 517369 914941 224440 340846 018932 650809 376948 638828 120650 370655 > 4¹³⁹ [i]