MinT - Best Known (125−45, 125, s)-Nets in Base 4

Best Known (125−45, 125, s)-Nets in Base 4

(125−45, 125, 139)-Net over F₄ — Constructive and digital

Digital (80, 125, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (57, 102, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 51, 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]

(125−45, 125, 280)-Net over F₄ — Digital

Digital (80, 125, 280)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−45, 125, 7449)-Net in Base 4 — Upper bound on s

There is no (80, 125, 7450)-net in base 4, because

1 times m-reduction [i] would yield (80, 124, 7450)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 453 066102 641765 611888 558542 960881 553739 320578 559123 787861 111123 790834 957836 > 4¹²⁴ [i]