MinT - Best Known (38, 38+107, s)-Nets in Base 4

Best Known (38, 38+107, s)-Nets in Base 4

(38, 38+107, 56)-Net over F₄ — Constructive and digital

Digital (38, 145, 56)-net over F₄, using

t-expansion [i] based on digital (33, 145, 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+107, 66)-Net over F₄ — Digital

Digital (38, 145, 66)-net over F₄, using

t-expansion [i] based on digital (37, 145, 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+107, 198)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 145, 199)-net over F₄, because

3 times m-reduction [i] would yield digital (38, 142, 199)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴², 199, F₄, 104) (dual of [199, 57, 105]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁴¹, 163, F₄, 104) (dual of [163, 22, 105]-code), but
      - constructionÂ Y1 [i] would yield
        OA(4¹⁴⁰, 151, S₄, 104), but
        2 times truncation [i] would yield 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]
        OA(4²², 163, S₄, 12), but
        the linear programming bound shows that MÂ â‰¥ 22822 886502 605724 057600 / 1273 153051 > 4²² [i]
    2. OA(4⁵⁷, 199, S₄, 36), but
      - discarding factors would yield OA(4⁵⁷, 195, S₄, 36), but
        the linear programming bound shows that MÂ â‰¥ 494 490067 436667 455875 598203 818411 591542 653847 684514 372082 636620 702806 966272 / 22309 972171 916724 959618 830756 395146 773299 > 4⁵⁷ [i]

(38, 38+107, 255)-Net in Base 4 — Upper bound on s

There is no (38, 145, 256)-net in base 4, because

1 times m-reduction [i] would yield (38, 144, 256)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 552 724342 739735 384585 163583 461979 462331 793806 695022 910459 059681 191039 433394 660082 426969 > 4¹⁴⁴ [i]