MinT - Best Known (112−37, 112, s)-Nets in Base 4

Best Known (112−37, 112, s)-Nets in Base 4

(112−37, 112, 195)-Net over F₄ — Constructive and digital

Digital (75, 112, 195)-net over F₄, using

4¹ times duplication [i] based on 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]

(112−37, 112, 196)-Net in Base 4 — Constructive

(75, 112, 196)-net in base 4, using

4² 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]

(112−37, 112, 353)-Net over F₄ — Digital

Digital (75, 112, 353)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−37, 112, 12978)-Net in Base 4 — Upper bound on s

There is no (75, 112, 12979)-net in base 4, because

1 times m-reduction [i] would yield (75, 111, 12979)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 743859 423813 886744 392522 258068 869359 757199 660237 083380 933633 834223 > 4¹¹¹ [i]