MinT - Best Known (149, 149+74, s)-Nets in Base 4

Best Known (149, 149+74, s)-Nets in Base 4

(149, 149+74, 195)-Net over F₄ — Constructive and digital

Digital (149, 223, 195)-net over F₄, using

4¹ times duplication [i] based on digital (148, 222, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 74, 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]

(149, 149+74, 240)-Net in Base 4 — Constructive

(149, 223, 240)-net in base 4, using

7 times m-reduction [i] based on (149, 230, 240)-net in base 4, using
- trace code for nets [i] based on (34, 115, 120)-net in base 16, using
  - base change [i] based on digital (11, 92, 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]

(149, 149+74, 615)-Net over F₄ — Digital

Digital (149, 223, 615)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(149, 149+74, 20739)-Net in Base 4 — Upper bound on s

There is no (149, 223, 20740)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 030712 645750 778339 822576 202528 624359 294229 752972 841165 295727 713289 306719 734875 607456 605695 164306 113206 151459 698961 457025 771153 122368 > 4²²³ [i]