MinT - Best Known (212−167, 212, s)-Nets in Base 4

Best Known (212−167, 212, s)-Nets in Base 4

(212−167, 212, 56)-Net over F₄ — Constructive and digital

Digital (45, 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]

(212−167, 212, 80)-Net over F₄ — Digital

Digital (45, 212, 80)-net over F₄, using

net from sequence [i] based on digital (45, 79)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 45 and N(F) ≥ 80, using
  - Drinfeld modules of rank 1 [i]

(212−167, 212, 190)-Net over F₄ — Upper bound on s (digital)

There is no digital (45, 212, 191)-net over F₄, because

31 times m-reduction [i] would yield digital (45, 181, 191)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁸¹, 191, F₄, 136) (dual of [191, 10, 137]-code), but
  - residual code [i] would yield linear OA(4⁴⁵, 54, F₄, 34) (dual of [54, 9, 35]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(212−167, 212, 194)-Net in Base 4 — Upper bound on s

There is no (45, 212, 195)-net in base 4, because

21 times m-reduction [i] would yield (45, 191, 195)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁹¹, 195, S₄, 146), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 630 432099 142311 667396 464641 602297 820881 275828 327447 146687 172694 467931 548343 955369 782628 260078 158650 252906 047844 909056 / 49 > 4¹⁹¹ [i]