MinT - Best Known (193−49, 193, s)-Nets in Base 4

Best Known (193−49, 193, s)-Nets in Base 4

(193−49, 193, 540)-Net over F₄ — Constructive and digital

Digital (144, 193, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (119, 168, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 56, 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]

(193−49, 193, 648)-Net in Base 4 — Constructive

(144, 193, 648)-net in base 4, using

4⁴ times duplication [i] based on (140, 189, 648)-net in base 4, using
- trace code for nets [i] based on (14, 63, 216)-net in base 64, using
  - base change [i] based on digital (5, 54, 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]

(193−49, 193, 1670)-Net over F₄ — Digital

Digital (144, 193, 1670)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−49, 193, 214132)-Net in Base 4 — Upper bound on s

There is no (144, 193, 214133)-net in base 4, because

1 times m-reduction [i] would yield (144, 192, 214133)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39 405784 518901 163539 473431 475997 108921 075959 263132 116215 671636 244513 038054 036821 631695 080507 640727 750715 930596 106844 > 4¹⁹² [i]