MinT - Best Known (209−75, 209, s)-Nets in Base 4

Best Known (209−75, 209, s)-Nets in Base 4

(209−75, 209, 157)-Net over F₄ — Constructive and digital

Digital (134, 209, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 47, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (87, 162, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 81, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(209−75, 209, 196)-Net in Base 4 — Constructive

(134, 209, 196)-net in base 4, using

t-expansion [i] based on (133, 209, 196)-net in base 4, using
- 1 times m-reduction [i] based on (133, 210, 196)-net in base 4, using
  - trace code for nets [i] based on (28, 105, 98)-net in base 16, using
    - base change [i] based on digital (7, 84, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(209−75, 209, 441)-Net over F₄ — Digital

Digital (134, 209, 441)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−75, 209, 11809)-Net in Base 4 — Upper bound on s

There is no (134, 209, 11810)-net in base 4, because

1 times m-reduction [i] would yield (134, 208, 11810)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169501 879526 103996 830826 938918 361101 768150 026531 008694 643977 722117 901534 476596 168969 643258 731954 816442 244027 859887 575679 342720 > 4²⁰⁸ [i]