MinT - Best Known (229−76, 229, s)-Nets in Base 4

Best Known (229−76, 229, s)-Nets in Base 4

(229−76, 229, 195)-Net over F₄ — Constructive and digital

Digital (153, 229, 195)-net over F₄, using

4¹ times duplication [i] based on 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]

(229−76, 229, 240)-Net in Base 4 — Constructive

(153, 229, 240)-net in base 4, using

7 times m-reduction [i] based on (153, 236, 240)-net in base 4, using
- trace code for nets [i] based on (35, 118, 120)-net in base 16, using
  - 2 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]

(229−76, 229, 629)-Net over F₄ — Digital

Digital (153, 229, 629)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−76, 229, 21244)-Net in Base 4 — Upper bound on s

There is no (153, 229, 21245)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 744770 074843 593305 043259 685382 132858 341681 338414 345598 588386 573108 171581 907616 760197 106208 254078 263917 697366 262103 563937 742327 183874 694900 > 4²²⁹ [i]