MinT - Best Known (111−37, 111, s)-Nets in Base 4

Best Known (111−37, 111, s)-Nets in Base 4

(111−37, 111, 195)-Net over F₄ — Constructive and digital

Digital (74, 111, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 37, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(111−37, 111, 196)-Net in Base 4 — Constructive

(74, 111, 196)-net in base 4, using

4¹ times duplication [i] based on (73, 110, 196)-net in base 4, using
- trace code for nets [i] based on (18, 55, 98)-net in base 16, using
  - base change [i] based on digital (7, 44, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(111−37, 111, 339)-Net over F₄ — Digital

Digital (74, 111, 339)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−37, 111, 12015)-Net in Base 4 — Upper bound on s

There is no (74, 111, 12016)-net in base 4, because

1 times m-reduction [i] would yield (74, 110, 12016)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 686333 088503 388717 725756 582434 973969 185997 401625 470447 591257 830598 > 4¹¹⁰ [i]