MinT - Best Known (115, 115+69, s)-Nets in Base 4

Best Known (115, 115+69, s)-Nets in Base 4

(115, 115+69, 135)-Net over F₄ — Constructive and digital

Digital (115, 184, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 34, 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 (81, 150, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 75, 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]

(115, 115+69, 339)-Net over F₄ — Digital

Digital (115, 184, 339)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 115+69, 7821)-Net in Base 4 — Upper bound on s

There is no (115, 184, 7822)-net in base 4, because

1 times m-reduction [i] would yield (115, 183, 7822)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 150 363260 082598 574247 241554 931051 885542 930778 583865 827997 095741 926159 399742 101106 829165 457649 618993 700833 104360 > 4¹⁸³ [i]