MinT - Best Known (207−53, 207, s)-Nets in Base 4

Best Known (207−53, 207, s)-Nets in Base 4

(207−53, 207, 540)-Net over F₄ — Constructive and digital

Digital (154, 207, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 27, 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 (127, 180, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 60, 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]

(207−53, 207, 648)-Net in Base 4 — Constructive

(154, 207, 648)-net in base 4, using

t-expansion [i] based on (153, 207, 648)-net in base 4, using
- trace code for nets [i] based on (15, 69, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 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]

(207−53, 207, 1706)-Net over F₄ — Digital

Digital (154, 207, 1706)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−53, 207, 207154)-Net in Base 4 — Upper bound on s

There is no (154, 207, 207155)-net in base 4, because

1 times m-reduction [i] would yield (154, 206, 207155)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10577 138046 882711 270929 349857 280705 383632 200388 658402 189644 869583 025825 415627 155784 682314 540454 470097 381696 833801 619691 227852 > 4²⁰⁶ [i]