MinT - Best Known (137, 137+49, s)-Nets in Base 4

Best Known (137, 137+49, s)-Nets in Base 4

(137, 137+49, 531)-Net over F₄ — Constructive and digital

Digital (137, 186, 531)-net over F₄, using

9 times m-reduction [i] based on digital (137, 195, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 65, 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]

(137, 137+49, 576)-Net in Base 4 — Constructive

(137, 186, 576)-net in base 4, using

t-expansion [i] based on (136, 186, 576)-net in base 4, using
- trace code for nets [i] based on (12, 62, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
    - base change [i] based on digital (3, 54, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(137, 137+49, 1369)-Net over F₄ — Digital

Digital (137, 186, 1369)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+49, 142909)-Net in Base 4 — Upper bound on s

There is no (137, 186, 142910)-net in base 4, because

1 times m-reduction [i] would yield (137, 185, 142910)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2405 079049 148257 706334 940619 206162 354179 914973 537201 272880 752202 507930 873403 596499 278949 379793 801187 637178 684289 > 4¹⁸⁵ [i]