MinT - Best Known (108, 108+37, s)-Nets in Base 4

Best Known (108, 108+37, s)-Nets in Base 4

(108, 108+37, 531)-Net over F₄ — Constructive and digital

Digital (108, 145, 531)-net over F₄, using

t-expansion [i] based on digital (107, 145, 531)-net over F₄, using
- 5 times m-reduction [i] based on digital (107, 150, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 50, 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]

(108, 108+37, 576)-Net in Base 4 — Constructive

(108, 145, 576)-net in base 4, using

2 times m-reduction [i] based on (108, 147, 576)-net in base 4, using
- trace code for nets [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 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]

(108, 108+37, 1284)-Net over F₄ — Digital

Digital (108, 145, 1284)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(108, 108+37, 164987)-Net in Base 4 — Upper bound on s

There is no (108, 145, 164988)-net in base 4, because

1 times m-reduction [i] would yield (108, 144, 164988)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 497 360144 122443 175613 287023 171027 755334 918776 067540 109985 826319 975695 399158 360048 729755 > 4¹⁴⁴ [i]