MinT - Best Known (34, 34+80, s)-Nets in Base 4

Best Known (34, 34+80, s)-Nets in Base 4

(34, 34+80, 56)-Net over F₄ — Constructive and digital

Digital (34, 114, 56)-net over F₄, using

t-expansion [i] based on digital (33, 114, 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]

(34, 34+80, 65)-Net over F₄ — Digital

Digital (34, 114, 65)-net over F₄, using

t-expansion [i] based on digital (33, 114, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(34, 34+80, 228)-Net over F₄ — Upper bound on s (digital)

There is no digital (34, 114, 229)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹¹⁴, 229, F₄, 80) (dual of [229, 115, 81]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹¹³, 148, S₄, 80), but
    - the linear programming bound shows that MÂ â‰¥ 134390 122763 442072 744489 280693 386578 084491 173770 590049 483300 612364 941999 081453 236362 093214 564352 / 1168 599339 892305 226318 359375 > 4¹¹³ [i]
  2. linear OA(4¹¹⁵, 229, F₄, 81) (dual of [229, 114, 82]-code), but
    - discarding factors / shortening the dual code would yield linear OA(4¹¹⁵, 219, F₄, 81) (dual of [219, 104, 82]-code), but
      - constructionÂ Y1 [i] would yield
        OA(4¹¹⁴, 146, S₄, 81), but
        the linear programming bound shows that MÂ â‰¥ 7 594596 073360 304072 815867 736592 315637 386549 140598 576996 791410 829396 630589 236390 143121 686528 / 15336 256272 900153 260065 > 4¹¹⁴ [i]
        OA(4¹⁰⁴, 219, S₄, 73), but
        discarding factors would yield OA(4¹⁰⁴, 150, S₄, 73), but
        the linear programming bound shows that MÂ â‰¥ 32 028783 264027 932827 838658 827950 132017 443030 031565 779091 184319 519613 927802 631343 679332 066747 830675 439616 / 70932 376441 373736 310653 589982 582096 950125 > 4¹⁰⁴ [i]

(34, 34+80, 241)-Net in Base 4 — Upper bound on s

There is no (34, 114, 242)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 442 348609 258243 337956 411718 991650 597606 305952 333199 425088 628473 635422 > 4¹¹⁴ [i]