MinT - Best Known (112−31, 112, s)-Nets in Base 4

Best Known (112−31, 112, s)-Nets in Base 4

(112−31, 112, 384)-Net over F₄ — Constructive and digital

Digital (81, 112, 384)-net over F₄, using

2 times m-reduction [i] based on digital (81, 114, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 38, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(112−31, 112, 387)-Net in Base 4 — Constructive

(81, 112, 387)-net in base 4, using

4¹ times duplication [i] based on (80, 111, 387)-net in base 4, using
- trace code for nets [i] based on (6, 37, 129)-net in base 64, using
  - 5 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
    - base change [i] based on digital (0, 36, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]

(112−31, 112, 725)-Net over F₄ — Digital

Digital (81, 112, 725)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−31, 112, 61066)-Net in Base 4 — Upper bound on s

There is no (81, 112, 61067)-net in base 4, because

1 times m-reduction [i] would yield (81, 111, 61067)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 740589 552823 181026 339361 993229 140045 188211 193902 754972 280497 025792 > 4¹¹¹ [i]