MinT - Best Known (228−76, 228, s)-Nets in Base 4

Best Known (228−76, 228, s)-Nets in Base 4

(228−76, 228, 195)-Net over F₄ — Constructive and digital

Digital (152, 228, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 76, 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]

(228−76, 228, 240)-Net in Base 4 — Constructive

(152, 228, 240)-net in base 4, using

6 times m-reduction [i] based on (152, 234, 240)-net in base 4, using
- trace code for nets [i] based on (35, 117, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 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]

(228−76, 228, 617)-Net over F₄ — Digital

Digital (152, 228, 617)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−76, 228, 20482)-Net in Base 4 — Upper bound on s

There is no (152, 228, 20483)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186260 187538 121515 605913 821069 846710 451236 618913 970420 009466 232424 812513 422798 279425 516275 598193 147235 271536 814924 335999 669828 555778 708480 > 4²²⁸ [i]