MinT - Best Known (172−64, 172, s)-Nets in Base 4

Best Known (172−64, 172, s)-Nets in Base 4

(172−64, 172, 135)-Net over F₄ — Constructive and digital

Digital (108, 172, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 32, 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 (76, 140, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 70, 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]

(172−64, 172, 329)-Net over F₄ — Digital

Digital (108, 172, 329)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−64, 172, 7316)-Net in Base 4 — Upper bound on s

There is no (108, 172, 7317)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 967401 630094 132521 822088 821353 766878 914656 048265 418405 467012 581874 208519 830355 229536 538875 154167 995700 > 4¹⁷² [i]