MinT - Best Known (166−43, 166, s)-Nets in Base 4

Best Known (166−43, 166, s)-Nets in Base 4

(166−43, 166, 531)-Net over F₄ — Constructive and digital

Digital (123, 166, 531)-net over F₄, using

8 times m-reduction [i] based on digital (123, 174, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 58, 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]

(166−43, 166, 576)-Net in Base 4 — Constructive

(123, 166, 576)-net in base 4, using

2 times m-reduction [i] based on (123, 168, 576)-net in base 4, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(166−43, 166, 1340)-Net over F₄ — Digital

Digital (123, 166, 1340)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−43, 166, 155523)-Net in Base 4 — Upper bound on s

There is no (123, 166, 155524)-net in base 4, because

1 times m-reduction [i] would yield (123, 165, 155524)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2187 484668 445566 919774 428117 172816 986282 896298 512991 092417 962728 177862 908502 083800 763510 642532 385248 > 4¹⁶⁵ [i]