MinT - Best Known (179−45, 179, s)-Nets in Base 4

Best Known (179−45, 179, s)-Nets in Base 4

(179−45, 179, 540)-Net over F₄ — Constructive and digital

Digital (134, 179, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 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 (111, 156, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 52, 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]

(179−45, 179, 648)-Net in Base 4 — Constructive

(134, 179, 648)-net in base 4, using

1 times m-reduction [i] based on (134, 180, 648)-net in base 4, using
- trace code for nets [i] based on (14, 60, 216)-net in base 64, using
  - 3 times m-reduction [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]

(179−45, 179, 1640)-Net over F₄ — Digital

Digital (134, 179, 1640)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(179−45, 179, 224341)-Net in Base 4 — Upper bound on s

There is no (134, 179, 224342)-net in base 4, because

1 times m-reduction [i] would yield (134, 178, 224342)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146788 955410 365229 197605 424272 709086 568945 114365 203492 562265 219537 449133 080390 273392 527748 584987 069321 912760 > 4¹⁷⁸ [i]