MinT - Best Known (120−85, 120, s)-Nets in Base 4

Best Known (120−85, 120, s)-Nets in Base 4

(120−85, 120, 56)-Net over F₄ — Constructive and digital

Digital (35, 120, 56)-net over F₄, using

t-expansion [i] based on digital (33, 120, 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]

(120−85, 120, 65)-Net over F₄ — Digital

Digital (35, 120, 65)-net over F₄, using

t-expansion [i] based on digital (33, 120, 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]

(120−85, 120, 205)-Net over F₄ — Upper bound on s (digital)

There is no digital (35, 120, 206)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²⁰, 206, F₄, 85) (dual of [206, 86, 86]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹¹⁹, 146, S₄, 85), but
    - the linear programming bound shows that MÂ â‰¥ 1 306185 861782 156678 344154 323879 967306 788722 965824 745661 222897 819372 462131 226542 873931 415552 / 2 755682 499112 350745 > 4¹¹⁹ [i]
  2. OA(4⁸⁶, 206, 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]

(120−85, 120, 246)-Net in Base 4 — Upper bound on s

There is no (35, 120, 247)-net in base 4, because

1 times m-reduction [i] would yield (35, 119, 247)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 467705 001969 366568 639785 858946 487728 231356 033618 163217 541388 835241 047710 > 4¹¹⁹ [i]