MinT - Best Known (138, 138+51, s)-Nets in Base 4

Best Known (138, 138+51, s)-Nets in Base 4

(138, 138+51, 531)-Net over F₄ — Constructive and digital

Digital (138, 189, 531)-net over F₄, using

t-expansion [i] based on digital (137, 189, 531)-net over F₄, using
- 6 times m-reduction [i] based on 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]

(138, 138+51, 576)-Net in Base 4 — Constructive

(138, 189, 576)-net in base 4, using

trace code for nets [i] based on (12, 63, 192)-net in base 64, using
- base change [i] based on digital (3, 54, 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]

(138, 138+51, 1250)-Net over F₄ — Digital

Digital (138, 189, 1250)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+51, 114266)-Net in Base 4 — Upper bound on s

There is no (138, 189, 114267)-net in base 4, because

1 times m-reduction [i] would yield (138, 188, 114267)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 153935 236302 959639 073625 235306 277318 412214 619101 595271 740649 876975 370126 783661 313946 631294 365532 879931 804333 256554 > 4¹⁸⁸ [i]