MinT - Best Known (38, 38+109, s)-Nets in Base 4

Best Known (38, 38+109, s)-Nets in Base 4

(38, 38+109, 56)-Net over F₄ — Constructive and digital

Digital (38, 147, 56)-net over F₄, using

t-expansion [i] based on digital (33, 147, 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+109, 66)-Net over F₄ — Digital

Digital (38, 147, 66)-net over F₄, using

t-expansion [i] based on digital (37, 147, 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+109, 195)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 147, 196)-net over F₄, because

1 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+109, 254)-Net in Base 4 — Upper bound on s

There is no (38, 147, 255)-net in base 4, because

1 times m-reduction [i] would yield (38, 146, 255)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8721 573322 063068 315234 883463 151701 450336 460481 690889 385383 202019 837700 556125 190491 712391 > 4¹⁴⁶ [i]