MinT - Best Known (125−41, 125, s)-Nets in Base 4

Best Known (125−41, 125, s)-Nets in Base 4

(125−41, 125, 195)-Net over F₄ — Constructive and digital

Digital (84, 125, 195)-net over F₄, using

1 times m-reduction [i] based on digital (84, 126, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 42, 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]

(125−41, 125, 196)-Net in Base 4 — Constructive

(84, 125, 196)-net in base 4, using

3 times m-reduction [i] based on (84, 128, 196)-net in base 4, using
- trace code for nets [i] based on (20, 64, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (20, 65, 98)-net in base 16, using
    - base change [i] based on digital (7, 52, 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]

(125−41, 125, 395)-Net over F₄ — Digital

Digital (84, 125, 395)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−41, 125, 14944)-Net in Base 4 — Upper bound on s

There is no (84, 125, 14945)-net in base 4, because

1 times m-reduction [i] would yield (84, 124, 14945)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 452 486657 252547 832126 686391 121729 596324 758539 877494 303585 556076 204260 396664 > 4¹²⁴ [i]