MinT - Best Known (140−102, 140, s)-Nets in Base 4

Best Known (140−102, 140, s)-Nets in Base 4

(140−102, 140, 56)-Net over F₄ — Constructive and digital

Digital (38, 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]

(140−102, 140, 66)-Net over F₄ — Digital

Digital (38, 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]

(140−102, 140, 203)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 140, 204)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁰, 204, F₄, 102) (dual of [204, 64, 103]-code), but
- constructionÂ Y1 [i] would yield
  1. 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]
  2. OA(4⁶⁴, 204, S₄, 41), but
    - discarding factors would yield OA(4⁶⁴, 203, S₄, 41), but
      - the linear programming bound shows that MÂ â‰¥ 9237 828678 014661 888733 201438 074087 629177 850315 553208 766694 988543 356653 154675 278877 490717 655040 / 25 848321 162508 053807 967718 560003 483887 415061 992757 599441 > 4⁶⁴ [i]

(140−102, 140, 257)-Net in Base 4 — Upper bound on s

There is no (38, 140, 258)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 085496 784454 650940 891774 578589 568513 475950 799953 779995 615097 791884 489023 221247 683200 > 4¹⁴⁰ [i]