MinT - Best Known (37, 37+109, s)-Nets in Base 4

Best Known (37, 37+109, s)-Nets in Base 4

(37, 37+109, 56)-Net over F₄ — Constructive and digital

Digital (37, 146, 56)-net over F₄, using

t-expansion [i] based on digital (33, 146, 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]

(37, 37+109, 66)-Net over F₄ — Digital

Digital (37, 146, 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]

(37, 37+109, 162)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 146, 163)-net over F₄, because

7 times m-reduction [i] would yield digital (37, 139, 163)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁹, 163, F₄, 102) (dual of [163, 24, 103]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(4¹³⁸, 149, S₄, 102), but
      - the linear programming bound shows that MÂ â‰¥ 11 769278 720446 703333 267659 747956 551886 187319 922656 933070 157940 129170 842786 675855 465333 653504 / 85 907459 > 4¹³⁸ [i]
    2. OA(4²⁴, 163, S₄, 14), but
      - discarding factors would yield OA(4²⁴, 134, S₄, 14), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 292 368491 029312 > 4²⁴ [i]

(37, 37+109, 247)-Net in Base 4 — Upper bound on s

There is no (37, 146, 248)-net in base 4, because

1 times m-reduction [i] would yield (37, 145, 248)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2383 208633 191613 874714 018504 389056 035576 522135 912757 384557 292972 886981 654595 743413 766761 > 4¹⁴⁵ [i]