MinT - Best Known (186−68, 186, s)-Nets in Base 4

Best Known (186−68, 186, s)-Nets in Base 4

(186−68, 186, 145)-Net over F₄ — Constructive and digital

Digital (118, 186, 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 (80, 148, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 74, 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]

(186−68, 186, 152)-Net in Base 4 — Constructive

(118, 186, 152)-net in base 4, using

2 times m-reduction [i] based on (118, 188, 152)-net in base 4, using
- trace code for nets [i] based on (24, 94, 76)-net in base 16, using
  - 1 times m-reduction [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]

(186−68, 186, 374)-Net over F₄ — Digital

Digital (118, 186, 374)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−68, 186, 8843)-Net in Base 4 — Upper bound on s

There is no (118, 186, 8844)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9646 218173 826442 869947 903894 182199 227435 336622 779481 389288 436582 235609 034478 610286 516153 293176 543031 665181 945796 > 4¹⁸⁶ [i]