MinT - Best Known (171, 171+60, s)-Nets in Base 4

Best Known (171, 171+60, s)-Nets in Base 4

(171, 171+60, 536)-Net over F₄ — Constructive and digital

Digital (171, 231, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 30, 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 (141, 201, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 67, 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+60, 648)-Net in Base 4 — Constructive

(171, 231, 648)-net in base 4, using

t-expansion [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+60, 1761)-Net over F₄ — Digital

Digital (171, 231, 1761)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(171, 171+60, 173567)-Net in Base 4 — Upper bound on s

There is no (171, 231, 173568)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 909847 436574 719102 883037 311272 791211 617906 205978 215381 975818 558524 712188 504501 701602 219720 779983 657003 072029 054269 803203 430750 893608 308385 > 4²³¹ [i]