MinT - Best Known (126−42, 126, s)-Nets in Base 4

Best Known (126−42, 126, s)-Nets in Base 4

(126−42, 126, 195)-Net over F₄ — Constructive and digital

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]

(126−42, 126, 196)-Net in Base 4 — Constructive

(84, 126, 196)-net in base 4, using

2 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]

(126−42, 126, 374)-Net over F₄ — Digital

Digital (84, 126, 374)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−42, 126, 11833)-Net in Base 4 — Upper bound on s

There is no (84, 126, 11834)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7245 196224 743193 327503 570656 035774 802040 103610 889213 534283 311944 581569 831122 > 4¹²⁶ [i]