MinT - Best Known (212−172, 212, s)-Nets in Base 4

Best Known (212−172, 212, s)-Nets in Base 4

(212−172, 212, 56)-Net over F₄ — Constructive and digital

Digital (40, 212, 56)-net over F₄, using

t-expansion [i] based on digital (33, 212, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(212−172, 212, 75)-Net over F₄ — Digital

Digital (40, 212, 75)-net over F₄, using

net from sequence [i] based on digital (40, 74)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
  - Drinfeld modules of rank 1 [i]

(212−172, 212, 169)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 212, 170)-net over F₄, because

48 times m-reduction [i] would yield digital (40, 164, 170)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶⁴, 170, F₄, 124) (dual of [170, 6, 125]-code), but
  - residual code [i] would yield linear OA(4⁴⁰, 45, F₄, 31) (dual of [45, 5, 32]-code), but
    - â€œLizâ€ bound on codes from Brouwerâ€™s database [i]

(212−172, 212, 174)-Net in Base 4 — Upper bound on s

There is no (40, 212, 175)-net in base 4, because

41 times m-reduction [i] would yield (40, 171, 175)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁷¹, 175, S₄, 131), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 143 343663 499379 469475 676305 956380 433799 785311 823017 570233 599302 461682 679755 530300 504376 159569 382855 409664 / 11 > 4¹⁷¹ [i]