MinT - Best Known (131, 131+77, s)-Nets in Base 4

Best Known (131, 131+77, s)-Nets in Base 4

(131, 131+77, 145)-Net over F₄ — Constructive and digital

Digital (131, 208, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 42, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (89, 166, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 83, 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]

(131, 131+77, 152)-Net in Base 4 — Constructive

(131, 208, 152)-net in base 4, using

2 times m-reduction [i] based on (131, 210, 152)-net in base 4, using
- trace code for nets [i] based on (26, 105, 76)-net in base 16, using
  - base change [i] based on digital (5, 84, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(131, 131+77, 393)-Net over F₄ — Digital

Digital (131, 208, 393)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+77, 9503)-Net in Base 4 — Upper bound on s

There is no (131, 208, 9504)-net in base 4, because

1 times m-reduction [i] would yield (131, 207, 9504)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42315 031240 086345 961470 453066 612591 261260 229451 063081 291629 997922 074558 450757 757499 258346 313546 653871 775762 622433 254307 417448 > 4²⁰⁷ [i]