MinT - Best Known (137, 137+75, s)-Nets in Base 4

Best Known (137, 137+75, s)-Nets in Base 4

(137, 137+75, 160)-Net over F₄ — Constructive and digital

Digital (137, 212, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 50, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [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]

(137, 137+75, 208)-Net in Base 4 — Constructive

(137, 212, 208)-net in base 4, using

4² times duplication [i] based on (135, 210, 208)-net in base 4, using
- trace code for nets [i] based on (30, 105, 104)-net in base 16, using
  - base change [i] based on digital (9, 84, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(137, 137+75, 469)-Net over F₄ — Digital

Digital (137, 212, 469)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+75, 13217)-Net in Base 4 — Upper bound on s

There is no (137, 212, 13218)-net in base 4, because

1 times m-reduction [i] would yield (137, 211, 13218)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 831370 152033 292864 736832 601974 409255 069933 384286 454859 192016 338696 605728 105077 087274 019577 188175 011592 777330 899574 154929 260000 > 4²¹¹ [i]