MinT - Best Known (188−69, 188, s)-Nets in Base 4

Best Known (188−69, 188, s)-Nets in Base 4

(188−69, 188, 145)-Net over F₄ — Constructive and digital

Digital (119, 188, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 38, 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 (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]

(188−69, 188, 152)-Net in Base 4 — Constructive

(119, 188, 152)-net in base 4, using

2 times m-reduction [i] based on (119, 190, 152)-net in base 4, using
- trace code for nets [i] based on (24, 95, 76)-net in base 16, using
  - base change [i] based on digital (5, 76, 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]

(188−69, 188, 372)-Net over F₄ — Digital

Digital (119, 188, 372)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−69, 188, 9212)-Net in Base 4 — Upper bound on s

There is no (119, 188, 9213)-net in base 4, because

1 times m-reduction [i] would yield (119, 187, 9213)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38555 414406 629088 210780 785611 077556 920020 162161 427043 843976 428473 514902 581197 545551 950457 680715 889693 947609 965850 > 4¹⁸⁷ [i]