MinT - Best Known (177, 177+62, s)-Nets in Base 4

Best Known (177, 177+62, s)-Nets in Base 4

(177, 177+62, 540)-Net over F₄ — Constructive and digital

Digital (177, 239, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 32, 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 (145, 207, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 69, 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]

(177, 177+62, 648)-Net in Base 4 — Constructive

(177, 239, 648)-net in base 4, using

1 times m-reduction [i] based on (177, 240, 648)-net in base 4, using
- trace code for nets [i] based on (17, 80, 216)-net in base 64, using
  - 4 times m-reduction [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]

(177, 177+62, 1825)-Net over F₄ — Digital

Digital (177, 239, 1825)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(177, 177+62, 181359)-Net in Base 4 — Upper bound on s

There is no (177, 239, 181360)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 780556 271901 652664 946304 765409 557465 257365 635225 489361 655596 223436 366260 904436 008197 911364 284472 821209 132895 076527 518976 563581 612668 028234 812002 > 4²³⁹ [i]