MinT - Best Known (197−50, 197, s)-Nets in Base 4

Best Known (197−50, 197, s)-Nets in Base 4

(197−50, 197, 540)-Net over F₄ — Constructive and digital

Digital (147, 197, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 26, 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 (121, 171, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 57, 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]

(197−50, 197, 648)-Net in Base 4 — Constructive

(147, 197, 648)-net in base 4, using

1 times m-reduction [i] based on (147, 198, 648)-net in base 4, using
- trace code for nets [i] based on (15, 66, 216)-net in base 64, using
  - 4 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]

(197−50, 197, 1702)-Net over F₄ — Digital

Digital (147, 197, 1702)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−50, 197, 188230)-Net in Base 4 — Upper bound on s

There is no (147, 197, 188231)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40349 407534 583549 023463 272805 203245 916358 128563 341810 146310 987293 664106 838748 766562 045581 154778 648283 353912 625619 390096 > 4¹⁹⁷ [i]