MinT - Best Known (157, 157+54, s)-Nets in Base 4

Best Known (157, 157+54, s)-Nets in Base 4

(157, 157+54, 540)-Net over F₄ — Constructive and digital

Digital (157, 211, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 28, 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 (129, 183, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 61, 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]

(157, 157+54, 648)-Net in Base 4 — Constructive

(157, 211, 648)-net in base 4, using

4¹ times duplication [i] based on (156, 210, 648)-net in base 4, using
- t-expansion [i] based on (155, 210, 648)-net in base 4, using
  - trace code for nets [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]

(157, 157+54, 1738)-Net over F₄ — Digital

Digital (157, 211, 1738)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+54, 184595)-Net in Base 4 — Upper bound on s

There is no (157, 211, 184596)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 831538 625501 828965 968501 984874 274292 909982 427892 354865 114854 704191 343777 328441 347706 795519 005872 795670 923554 042396 888704 389728 > 4²¹¹ [i]