MinT - Best Known (187, 187+65, s)-Nets in Base 4

Best Known (187, 187+65, s)-Nets in Base 4

(187, 187+65, 546)-Net over F₄ — Constructive and digital

Digital (187, 252, 546)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 36, 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 (151, 216, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 72, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(187, 187+65, 648)-Net in Base 4 — Constructive

(187, 252, 648)-net in base 4, using

t-expansion [i] based on (185, 252, 648)-net in base 4, using
- trace code for nets [i] based on (17, 84, 216)-net in base 64, using
  - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(187, 187+65, 1963)-Net over F₄ — Digital

Digital (187, 252, 1963)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(187, 187+65, 224965)-Net in Base 4 — Upper bound on s

There is no (187, 252, 224966)-net in base 4, because

1 times m-reduction [i] would yield (187, 251, 224966)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 093780 812906 635052 211352 357533 081404 860942 707501 356558 584880 471171 969439 061329 758440 186794 474831 539357 047067 752250 669620 756147 318270 705953 123062 497794 > 4²⁵¹ [i]