MinT - Best Known (167, 167+58, s)-Nets in Base 4

Best Known (167, 167+58, s)-Nets in Base 4

(167, 167+58, 540)-Net over F₄ — Constructive and digital

Digital (167, 225, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 30, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [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]

(167, 167+58, 648)-Net in Base 4 — Constructive

(167, 225, 648)-net in base 4, using

t-expansion [i] based on (166, 225, 648)-net in base 4, using
- trace code for nets [i] based on (16, 75, 216)-net in base 64, using
  - 2 times m-reduction [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]

(167, 167+58, 1780)-Net over F₄ — Digital

Digital (167, 225, 1780)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(167, 167+58, 182431)-Net in Base 4 — Upper bound on s

There is no (167, 225, 182432)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2907 641192 814989 036977 238637 308792 445490 907982 299562 659757 646515 846004 429247 459909 274531 888797 421093 477471 526587 057414 161506 118867 053735 > 4²²⁵ [i]