MinT - Best Known (177, 177+61, s)-Nets in Base 4

Best Known (177, 177+61, s)-Nets in Base 4

(177, 177+61, 546)-Net over F₄ — Constructive and digital

Digital (177, 238, 546)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 34, 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 (143, 204, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 68, 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+61, 648)-Net in Base 4 — Constructive

(177, 238, 648)-net in base 4, using

2 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+61, 1919)-Net over F₄ — Digital

Digital (177, 238, 1919)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(177, 177+61, 229031)-Net in Base 4 — Upper bound on s

There is no (177, 238, 229032)-net in base 4, because

1 times m-reduction [i] would yield (177, 237, 229032)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48781 608994 689154 834916 661261 567728 764195 253154 402401 027963 639437 564032 132054 743912 901771 190899 104464 318169 690521 501581 086547 330788 007365 077324 > 4²³⁷ [i]