MinT - Best Known (250−89, 250, s)-Nets in Base 4

Best Known (250−89, 250, s)-Nets in Base 4

(250−89, 250, 200)-Net over F₄ — Constructive and digital

Digital (161, 250, 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]

(250−89, 250, 240)-Net in Base 4 — Constructive

(161, 250, 240)-net in base 4, using

trace code for nets [i] based on (36, 125, 120)-net in base 16, using
- base change [i] based on digital (11, 100, 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]

(250−89, 250, 531)-Net over F₄ — Digital

Digital (161, 250, 531)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−89, 250, 14650)-Net in Base 4 — Upper bound on s

There is no (161, 250, 14651)-net in base 4, because

1 times m-reduction [i] would yield (161, 249, 14651)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 820284 587549 062726 976233 947721 225131 253393 915816 749092 342069 756031 690271 695090 194842 294338 283014 536321 932756 597321 330808 832667 750134 754206 857578 985308 > 4²⁴⁹ [i]