MinT - Best Known (113, 113+39, s)-Nets in Base 4

Best Known (113, 113+39, s)-Nets in Base 4

(113, 113+39, 531)-Net over F₄ — Constructive and digital

Digital (113, 152, 531)-net over F₄, using

7 times m-reduction [i] based on digital (113, 159, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 53, 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]

(113, 113+39, 576)-Net in Base 4 — Constructive

(113, 152, 576)-net in base 4, using

1 times m-reduction [i] based on (113, 153, 576)-net in base 4, using
- trace code for nets [i] based on (11, 51, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
    - base change [i] based on digital (3, 48, 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]

(113, 113+39, 1301)-Net over F₄ — Digital

Digital (113, 152, 1301)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(113, 113+39, 161007)-Net in Base 4 — Upper bound on s

There is no (113, 152, 161008)-net in base 4, because

1 times m-reduction [i] would yield (113, 151, 161008)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 148855 764363 477315 557776 933741 147807 887687 099586 552692 411276 107436 503065 461881 689550 126918 > 4¹⁵¹ [i]