MinT - Best Known (171, 171+59, s)-Nets in Base 4

Best Known (171, 171+59, s)-Nets in Base 4

(171, 171+59, 545)-Net over F₄ — Constructive and digital

Digital (171, 230, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 32, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (139, 198, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 66, 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]

(171, 171+59, 648)-Net in Base 4 — Constructive

(171, 230, 648)-net in base 4, using

t-expansion [i] based on (170, 230, 648)-net in base 4, using
- 1 times m-reduction [i] based on (170, 231, 648)-net in base 4, using
  - trace code for nets [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 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]

(171, 171+59, 1855)-Net over F₄ — Digital

Digital (171, 230, 1855)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(171, 171+59, 220877)-Net in Base 4 — Upper bound on s

There is no (171, 230, 220878)-net in base 4, because

1 times m-reduction [i] would yield (171, 229, 220878)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 744300 473746 534348 662558 400719 797673 583006 216734 576080 410559 822367 466062 176415 058090 957447 894595 382482 457808 731179 924146 277463 822695 092324 > 4²²⁹ [i]