MinT - Best Known (203−52, 203, s)-Nets in Base 4

Best Known (203−52, 203, s)-Nets in Base 4

(203−52, 203, 536)-Net over F₄ — Constructive and digital

Digital (151, 203, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 26, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (125, 177, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 59, 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]

(203−52, 203, 648)-Net in Base 4 — Constructive

(151, 203, 648)-net in base 4, using

1 times m-reduction [i] based on (151, 204, 648)-net in base 4, using
- trace code for nets [i] based on (15, 68, 216)-net in base 64, using
  - 2 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]

(203−52, 203, 1674)-Net over F₄ — Digital

Digital (151, 203, 1674)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(203−52, 203, 176530)-Net in Base 4 — Upper bound on s

There is no (151, 203, 176531)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 284784 549524 082832 985688 208104 765478 671180 168023 337570 835612 198811 733755 236360 917097 290933 850961 813307 127442 022589 992688 > 4²⁰³ [i]