MinT - Best Known (171, 171+58, s)-Nets in Base 4

Best Known (171, 171+58, s)-Nets in Base 4

(171, 171+58, 548)-Net over F₄ — Constructive and digital

Digital (171, 229, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 34, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (137, 195, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 65, 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+58, 648)-Net in Base 4 — Constructive

(171, 229, 648)-net in base 4, using

t-expansion [i] based on (170, 229, 648)-net in base 4, using
- 2 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+58, 1959)-Net over F₄ — Digital

Digital (171, 229, 1959)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(171, 171+58, 220877)-Net in Base 4 — Upper bound on s

There is no (171, 229, 220878)-net in base 4, because

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]