MinT - Best Known (125, 125+73, s)-Nets in Base 4

Best Known (125, 125+73, s)-Nets in Base 4

(125, 125+73, 145)-Net over F₄ — Constructive and digital

Digital (125, 198, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 40, 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 (85, 158, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 79, 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, 125+73, 152)-Net in Base 4 — Constructive

(125, 198, 152)-net in base 4, using

2 times m-reduction [i] based on (125, 200, 152)-net in base 4, using
- trace code for nets [i] based on (25, 100, 76)-net in base 16, using
  - base change [i] based on digital (5, 80, 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]

(125, 125+73, 383)-Net over F₄ — Digital

Digital (125, 198, 383)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+73, 9350)-Net in Base 4 — Upper bound on s

There is no (125, 198, 9351)-net in base 4, because

1 times m-reduction [i] would yield (125, 197, 9351)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40355 835537 723262 160278 292825 606124 731150 721111 925623 286372 012649 058711 433541 918092 880689 741768 074177 426156 760479 012964 > 4¹⁹⁷ [i]