MinT - Best Known (176, 176+62, s)-Nets in Base 4

Best Known (176, 176+62, s)-Nets in Base 4

(176, 176+62, 536)-Net over F₄ — Constructive and digital

Digital (176, 238, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 31, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [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]

(176, 176+62, 648)-Net in Base 4 — Constructive

(176, 238, 648)-net in base 4, using

4¹ times duplication [i] based on (175, 237, 648)-net in base 4, using
- trace code for nets [i] based on (17, 79, 216)-net in base 64, using
  - 5 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]

(176, 176+62, 1785)-Net over F₄ — Digital

Digital (176, 238, 1785)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(176, 176+62, 173426)-Net in Base 4 — Upper bound on s

There is no (176, 238, 173427)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195128 882789 624723 633567 937649 194392 600138 599388 850086 761287 867802 855927 361584 314510 172415 428114 836226 649695 840867 810548 206974 184133 167787 431680 > 4²³⁸ [i]