MinT - Best Known (198−66, 198, s)-Nets in Base 4

Best Known (198−66, 198, s)-Nets in Base 4

(198−66, 198, 195)-Net over F₄ — Constructive and digital

Digital (132, 198, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 66, 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]

(198−66, 198, 240)-Net in Base 4 — Constructive

(132, 198, 240)-net in base 4, using

t-expansion [i] based on (131, 198, 240)-net in base 4, using
- 2 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
  - trace code for nets [i] based on (31, 100, 120)-net in base 16, using
    - base change [i] based on digital (11, 80, 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]

(198−66, 198, 545)-Net over F₄ — Digital

Digital (132, 198, 545)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−66, 198, 17944)-Net in Base 4 — Upper bound on s

There is no (132, 198, 17945)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161430 217562 680170 827702 168588 910264 163722 129376 796122 708504 919989 757569 949845 541941 544396 634835 808449 511835 185976 669760 > 4¹⁹⁸ [i]