MinT - Best Known (163−128, 163, s)-Nets in Base 4

Best Known (163−128, 163, s)-Nets in Base 4

(163−128, 163, 56)-Net over F₄ — Constructive and digital

Digital (35, 163, 56)-net over F₄, using

t-expansion [i] based on digital (33, 163, 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]

(163−128, 163, 65)-Net over F₄ — Digital

Digital (35, 163, 65)-net over F₄, using

t-expansion [i] based on digital (33, 163, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(163−128, 163, 144)-Net in Base 4 — Upper bound on s

There is no (35, 163, 145)-net in base 4, because

32 times m-reduction [i] would yield (35, 131, 145)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹³¹, 145, S₄, 96), but
  - the linear programming bound shows that MÂ â‰¥ 4 521509 394595 294317 880132 762660 256020 199722 572909 200327 946519 584732 236773 399422 238875 189248 / 570286 240909 > 4¹³¹ [i]