MinT - Best Known (111−31, 111, s)-Nets in Base 4

Best Known (111−31, 111, s)-Nets in Base 4

(111−31, 111, 384)-Net over F₄ — Constructive and digital

Digital (80, 111, 384)-net over F₄, using

t-expansion [i] based on digital (79, 111, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 37, 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]

(111−31, 111, 387)-Net in Base 4 — Constructive

(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]

(111−31, 111, 693)-Net over F₄ — Digital

Digital (80, 111, 693)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−31, 111, 55674)-Net in Base 4 — Upper bound on s

There is no (80, 111, 55675)-net in base 4, because

1 times m-reduction [i] would yield (80, 110, 55675)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 685102 883141 356548 842219 827267 675654 040739 187086 323925 546956 752416 > 4¹¹⁰ [i]