MinT - Best Known (39, 140, s)-Nets in Base 4

Best Known (39, 140, s)-Nets in Base 4

(39, 140, 56)-Net over F₄ — Constructive and digital

Digital (39, 140, 56)-net over F₄, using

t-expansion [i] based on digital (33, 140, 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]

(39, 140, 66)-Net over F₄ — Digital

Digital (39, 140, 66)-net over F₄, using

t-expansion [i] based on digital (37, 140, 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]

(39, 140, 244)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 140, 245)-net over F₄, because

1 times m-reduction [i] would yield digital (39, 139, 245)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁹, 245, F₄, 100) (dual of [245, 106, 101]-code), but
  - residual code [i] would yield OA(4³⁹, 144, S₄, 25), but
    - the linear programming bound shows that MÂ â‰¥ 451 194621 572690 925216 028943 220812 151783 424000 / 1452 353674 315137 592771 > 4³⁹ [i]

(39, 140, 267)-Net in Base 4 — Upper bound on s

There is no (39, 140, 268)-net in base 4, because

1 times m-reduction [i] would yield (39, 139, 268)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 553968 608355 777854 709339 941545 009172 514678 999713 657800 132296 322460 314547 148504 325456 > 4¹³⁹ [i]