MinT - Best Known (188, 188+65, s)-Nets in Base 4

Best Known (188, 188+65, s)-Nets in Base 4

(188, 188+65, 548)-Net over F₄ — Constructive and digital

Digital (188, 253, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 37, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, 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]

(188, 188+65, 648)-Net in Base 4 — Constructive

(188, 253, 648)-net in base 4, using

4¹ times duplication [i] based on (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]

(188, 188+65, 2005)-Net over F₄ — Digital

Digital (188, 253, 2005)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(188, 188+65, 234927)-Net in Base 4 — Upper bound on s

There is no (188, 253, 234928)-net in base 4, because

1 times m-reduction [i] would yield (188, 252, 234928)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 380444 036670 373486 380277 145180 376812 474997 930167 635692 436140 831954 010168 912143 375859 184677 707620 437580 399252 265822 716646 000268 554337 903175 184051 634905 > 4²⁵² [i]