MinT - Best Known (38, 38+104, s)-Nets in Base 4

Best Known (38, 38+104, s)-Nets in Base 4

(38, 38+104, 56)-Net over F₄ — Constructive and digital

Digital (38, 142, 56)-net over F₄, using

t-expansion [i] based on digital (33, 142, 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+104, 66)-Net over F₄ — Digital

Digital (38, 142, 66)-net over F₄, using

t-expansion [i] based on digital (37, 142, 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+104, 198)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 142, 199)-net over F₄, because

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
      1. 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]
      2. 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+104, 256)-Net in Base 4 — Upper bound on s

There is no (38, 142, 257)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34 310653 274500 056749 199773 507551 670278 936871 414696 053982 060394 202846 382748 555446 422720 > 4¹⁴² [i]