MinT - Best Known (255−89, 255, s)-Nets in Base 4

Best Known (255−89, 255, s)-Nets in Base 4

(255−89, 255, 200)-Net over F₄ — Constructive and digital

Digital (166, 255, 200)-net over F₄, using

t-expansion [i] based on digital (161, 255, 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]

(255−89, 255, 240)-Net in Base 4 — Constructive

(166, 255, 240)-net in base 4, using

3 times m-reduction [i] based on (166, 258, 240)-net in base 4, using
- trace code for nets [i] based on (37, 129, 120)-net in base 16, using
  - 1 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]

(255−89, 255, 580)-Net over F₄ — Digital

Digital (166, 255, 580)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−89, 255, 17155)-Net in Base 4 — Upper bound on s

There is no (166, 255, 17156)-net in base 4, because

1 times m-reduction [i] would yield (166, 254, 17156)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 838 012431 516407 560707 551356 504780 794270 733351 440708 274469 522499 458038 785073 948350 869109 475615 895118 296419 363137 020795 064981 565315 310522 561343 628770 031936 > 4²⁵⁴ [i]