MinT - Best Known (131, 131+44, s)-Nets in Base 4

Best Known (131, 131+44, s)-Nets in Base 4

(131, 131+44, 536)-Net over F₄ — Constructive and digital

Digital (131, 175, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 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 (109, 153, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 51, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(131, 131+44, 648)-Net in Base 4 — Constructive

(131, 175, 648)-net in base 4, using

4¹ times duplication [i] based on (130, 174, 648)-net in base 4, using
- trace code for nets [i] based on (14, 58, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (14, 63, 216)-net in base 64, using
    - base change [i] based on digital (5, 54, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(131, 131+44, 1608)-Net over F₄ — Digital

Digital (131, 175, 1608)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+44, 185696)-Net in Base 4 — Upper bound on s

There is no (131, 175, 185697)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2293 644052 593234 882627 196718 118563 687854 054190 762395 601729 940523 765268 338535 306644 942720 894385 482226 655760 > 4¹⁷⁵ [i]