MinT - Best Known (117−83, 117, s)-Nets in Base 4

Best Known (117−83, 117, s)-Nets in Base 4

(117−83, 117, 56)-Net over F₄ — Constructive and digital

Digital (34, 117, 56)-net over F₄, using

t-expansion [i] based on digital (33, 117, 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]

(117−83, 117, 65)-Net over F₄ — Digital

Digital (34, 117, 65)-net over F₄, using

t-expansion [i] based on digital (33, 117, 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]

(117−83, 117, 202)-Net over F₄ — Upper bound on s (digital)

There is no digital (34, 117, 203)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹¹⁷, 203, F₄, 83) (dual of [203, 86, 84]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹¹⁶, 143, S₄, 83), but
    - the linear programming bound shows that MÂ â‰¥ 247019 501721 767325 521677 985124 228608 567579 718666 161103 517597 971229 442510 874624 631519 576064 / 26 250606 256753 373495 > 4¹¹⁶ [i]
  2. OA(4⁸⁶, 203, S₄, 60), but
    - discarding factors would yield OA(4⁸⁶, 145, S₄, 60), but
      - the linear programming bound shows that MÂ â‰¥ 70 960269 863126 606518 991667 215730 563206 075021 389069 425781 018295 735734 868068 159984 954116 355960 140231 669388 460318 751919 101866 886859 928826 023614 873600 / 11532 795767 709720 554647 969916 716816 544438 820799 421172 775769 384715 477815 805045 795696 528849 736837 > 4⁸⁶ [i]

(117−83, 117, 239)-Net in Base 4 — Upper bound on s

There is no (34, 117, 240)-net in base 4, because

1 times m-reduction [i] would yield (34, 116, 240)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7340 566728 744543 186578 249241 522925 991484 586465 678236 031165 098079 226305 > 4¹¹⁶ [i]