MinT - Best Known (254−89, 254, s)-Nets in Base 4

Best Known (254−89, 254, s)-Nets in Base 4

(254−89, 254, 200)-Net over F₄ — Constructive and digital

Digital (165, 254, 200)-net over F₄, using

t-expansion [i] based on digital (161, 254, 200)-net over F₄, using
- net from sequence [i] based on digital (161, 199)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
    - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(254−89, 254, 240)-Net in Base 4 — Constructive

(165, 254, 240)-net in base 4, using

2 times m-reduction [i] based on (165, 256, 240)-net in base 4, using
- trace code for nets [i] based on (37, 128, 120)-net in base 16, using
  - 2 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
    - base change [i] based on digital (11, 104, 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]

(254−89, 254, 570)-Net over F₄ — Digital

Digital (165, 254, 570)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−89, 254, 16622)-Net in Base 4 — Upper bound on s

There is no (165, 254, 16623)-net in base 4, because

1 times m-reduction [i] would yield (165, 253, 16623)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 209 626130 129854 652299 988607 788443 742712 410510 910867 480050 818759 934695 246796 401251 421256 480621 355135 119432 973185 517600 650992 870212 266902 145416 805592 473521 > 4²⁵³ [i]