MinT - Best Known (138−48, 138, s)-Nets in Base 4

Best Known (138−48, 138, s)-Nets in Base 4

(138−48, 138, 147)-Net over F₄ — Constructive and digital

Digital (90, 138, 147)-net over F₄, using

1 times m-reduction [i] based on digital (90, 139, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 29, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (61, 110, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 55, 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]

(138−48, 138, 196)-Net in Base 4 — Constructive

(90, 138, 196)-net in base 4, using

trace code for nets [i] based on (21, 69, 98)-net in base 16, using
- 1 times m-reduction [i] based on (21, 70, 98)-net in base 16, using
  - base change [i] based on digital (7, 56, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(138−48, 138, 344)-Net over F₄ — Digital

Digital (90, 138, 344)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−48, 138, 9444)-Net in Base 4 — Upper bound on s

There is no (90, 138, 9445)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 121487 505249 196321 765207 031146 464173 116782 011290 425436 580370 003062 464562 813087 084044 > 4¹³⁸ [i]