MinT - Best Known (112, 167, s)-Nets in Base 4

Best Known (112, 167, s)-Nets in Base 4

(112, 167, 195)-Net over F₄ — Constructive and digital

Digital (112, 167, 195)-net over F₄, using

1 times m-reduction [i] based on digital (112, 168, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 56, 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, 167, 240)-Net in Base 4 — Constructive

(112, 167, 240)-net in base 4, using

1 times m-reduction [i] based on (112, 168, 240)-net in base 4, using
- trace code for nets [i] based on (28, 84, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (28, 85, 120)-net in base 16, using
    - base change [i] based on digital (11, 68, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(112, 167, 493)-Net over F₄ — Digital

Digital (112, 167, 493)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(112, 167, 18294)-Net in Base 4 — Upper bound on s

There is no (112, 167, 18295)-net in base 4, because

1 times m-reduction [i] would yield (112, 166, 18295)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8756 930533 909864 200359 737240 552670 536351 032921 049410 913871 403756 585938 453204 183481 503785 785526 935712 > 4¹⁶⁶ [i]