MinT - Best Known (118, 118+71, s)-Nets in Base 4

Best Known (118, 118+71, s)-Nets in Base 4

(118, 118+71, 135)-Net over F₄ — Constructive and digital

Digital (118, 189, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 35, 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 (83, 154, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 77, 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]

(118, 118+71, 344)-Net over F₄ — Digital

Digital (118, 189, 344)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(118, 118+71, 7915)-Net in Base 4 — Upper bound on s

There is no (118, 189, 7916)-net in base 4, because

1 times m-reduction [i] would yield (118, 188, 7916)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 154022 531113 406103 310940 694841 318494 837064 066380 290066 955190 039209 666131 518970 842308 818934 366244 881352 505614 167685 > 4¹⁸⁸ [i]