MinT - Best Known (38, 38+110, s)-Nets in Base 4

Best Known (38, 38+110, s)-Nets in Base 4

(38, 38+110, 56)-Net over F₄ — Constructive and digital

Digital (38, 148, 56)-net over F₄, using

t-expansion [i] based on digital (33, 148, 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+110, 66)-Net over F₄ — Digital

Digital (38, 148, 66)-net over F₄, using

t-expansion [i] based on digital (37, 148, 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+110, 195)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 148, 196)-net over F₄, because

2 times m-reduction [i] would yield digital (38, 146, 196)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁶, 196, F₄, 108) (dual of [196, 50, 109]-code), but
  - residual code [i] would yield OA(4³⁸, 87, S₄, 27), but
    - the linear programming bound shows that MÂ â‰¥ 750676 473656 767244 506582 044594 315077 520225 769567 027200 / 9 306912 430727 497773 727968 244619 > 4³⁸ [i]

(38, 38+110, 253)-Net in Base 4 — Upper bound on s

There is no (38, 148, 254)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 134843 032509 645894 630129 217550 627572 239441 528355 232947 142833 534039 788558 938845 227644 374924 > 4¹⁴⁸ [i]