MinT - Best Known (119−84, 119, s)-Nets in Base 4

Best Known (119−84, 119, s)-Nets in Base 4

(119−84, 119, 56)-Net over F₄ — Constructive and digital

Digital (35, 119, 56)-net over F₄, using

t-expansion [i] based on digital (33, 119, 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]

(119−84, 119, 65)-Net over F₄ — Digital

Digital (35, 119, 65)-net over F₄, using

t-expansion [i] based on digital (33, 119, 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]

(119−84, 119, 215)-Net over F₄ — Upper bound on s (digital)

There is no digital (35, 119, 216)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹¹⁹, 216, F₄, 84) (dual of [216, 97, 85]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹¹⁸, 148, S₄, 84), but
    - the linear programming bound shows that MÂ â‰¥ 7301 270043 157677 378834 968797 352546 115167 116693 516048 447726 743341 396059 833062 672218 026684 710912 / 58934 105814 768105 796875 > 4¹¹⁸ [i]
  2. OA(4⁹⁷, 216, S₄, 68), but
    - discarding factors would yield OA(4⁹⁷, 147, S₄, 68), but
      - the linear programming bound shows that MÂ â‰¥ 53191 285542 209075 945504 006887 276132 854449 205165 228826 719771 521828 218168 917559 533017 381886 236463 273092 752842 577445 650432 / 2 089901 251760 098724 970145 084401 522230 137171 459432 895605 490625 > 4⁹⁷ [i]

(119−84, 119, 246)-Net in Base 4 — Upper bound on s

There is no (35, 119, 247)-net in base 4, because

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]