MinT - Best Known (36, 124, s)-Nets in Base 4

Best Known (36, 124, s)-Nets in Base 4

(36, 124, 56)-Net over F₄ — Constructive and digital

Digital (36, 124, 56)-net over F₄, using

t-expansion [i] based on digital (33, 124, 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]

(36, 124, 65)-Net over F₄ — Digital

Digital (36, 124, 65)-net over F₄, using

t-expansion [i] based on digital (33, 124, 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]

(36, 124, 209)-Net over F₄ — Upper bound on s (digital)

There is no digital (36, 124, 210)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²⁴, 210, F₄, 88) (dual of [210, 86, 89]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹²³, 150, S₄, 88), but
    - the linear programming bound shows that MÂ â‰¥ 22915 666113 439092 609099 444232 414491 754857 959622 104295 992957 494311 648680 329878 580928 609806 974976 / 136 168406 561048 662801 > 4¹²³ [i]
  2. OA(4⁸⁶, 210, 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]

(36, 124, 251)-Net in Base 4 — Upper bound on s

There is no (36, 124, 252)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 482 914408 740394 517158 583005 380335 490356 741743 841311 794512 345787 846610 298884 > 4¹²⁴ [i]