MinT - Best Known (112, 112+67, s)-Nets in Base 4

Best Known (112, 112+67, s)-Nets in Base 4

(112, 112+67, 135)-Net over F₄ — Constructive and digital

Digital (112, 179, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 33, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (79, 146, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 73, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(112, 112+67, 333)-Net over F₄ — Digital

Digital (112, 179, 333)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(112, 112+67, 7730)-Net in Base 4 — Upper bound on s

There is no (112, 179, 7731)-net in base 4, because

1 times m-reduction [i] would yield (112, 178, 7731)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 147228 453975 902760 677483 705545 123726 060771 837277 409651 873353 638099 379846 645195 040314 406325 031449 475922 718438 > 4¹⁷⁸ [i]