MinT - Best Known (38, 38+87, s)-Nets in Base 4

Best Known (38, 38+87, s)-Nets in Base 4

(38, 38+87, 56)-Net over F₄ — Constructive and digital

Digital (38, 125, 56)-net over F₄, using

t-expansion [i] based on digital (33, 125, 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]

(38, 38+87, 66)-Net over F₄ — Digital

Digital (38, 125, 66)-net over F₄, using

t-expansion [i] based on digital (37, 125, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

(38, 38+87, 272)-Net in Base 4 — Upper bound on s

There is no (38, 125, 273)-net in base 4, because

1 times m-reduction [i] would yield (38, 124, 273)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 471 662455 821194 487181 338912 785435 866273 943246 344542 717088 534343 409752 606080 > 4¹²⁴ [i]