MinT - Best Known (137, 137+50, s)-Nets in Base 4

Best Known (137, 137+50, s)-Nets in Base 4

(137, 137+50, 531)-Net over F₄ — Constructive and digital

Digital (137, 187, 531)-net over F₄, using

8 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+50, 576)-Net in Base 4 — Constructive

(137, 187, 576)-net in base 4, using

4¹ times duplication [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+50, 1289)-Net over F₄ — Digital

Digital (137, 187, 1289)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+50, 108101)-Net in Base 4 — Upper bound on s

There is no (137, 187, 108102)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38483 164448 587771 343008 063647 725778 246729 218213 313003 158845 695895 472594 397870 614354 491295 233862 350635 994287 902056 > 4¹⁸⁷ [i]