MinT - Best Known (131, 131+68, s)-Nets in Base 4

Best Known (131, 131+68, s)-Nets in Base 4

(131, 131+68, 163)-Net over F₄ — Constructive and digital

Digital (131, 199, 163)-net over F₄, using

t-expansion [i] based on digital (130, 199, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 49, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [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]

(131, 131+68, 240)-Net in Base 4 — Constructive

(131, 199, 240)-net in base 4, using

1 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
- trace code for nets [i] based on (31, 100, 120)-net in base 16, using
  - base change [i] based on digital (11, 80, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(131, 131+68, 500)-Net over F₄ — Digital

Digital (131, 199, 500)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+68, 15044)-Net in Base 4 — Upper bound on s

There is no (131, 199, 15045)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 646495 104238 417403 058917 657661 035385 387007 253639 168033 399894 121301 005522 028813 526030 900259 762175 488888 518984 200468 245676 > 4¹⁹⁹ [i]