MinT - Best Known (142, 142+48, s)-Nets in Base 4

Best Known (142, 142+48, s)-Nets in Base 4

(142, 142+48, 540)-Net over F₄ — Constructive and digital

Digital (142, 190, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 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 (117, 165, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 55, 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]

(142, 142+48, 648)-Net in Base 4 — Constructive

(142, 190, 648)-net in base 4, using

4¹ times duplication [i] based on (141, 189, 648)-net in base 4, using
- t-expansion [i] based on (140, 189, 648)-net in base 4, using
  - trace code for nets [i] based on (14, 63, 216)-net in base 64, using
    - base change [i] based on digital (5, 54, 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]

(142, 142+48, 1686)-Net over F₄ — Digital

Digital (142, 190, 1686)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(142, 142+48, 190767)-Net in Base 4 — Upper bound on s

There is no (142, 190, 190768)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 462648 422369 409763 695319 770001 915714 904294 527225 880722 489416 305211 258542 891410 317288 087159 086641 045737 673902 591614 > 4¹⁹⁰ [i]