MinT - Best Known (36, 128, s)-Nets in Base 4

Best Known (36, 128, s)-Nets in Base 4

(36, 128, 56)-Net over F₄ — Constructive and digital

Digital (36, 128, 56)-net over F₄, using

t-expansion [i] based on digital (33, 128, 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, 128, 65)-Net over F₄ — Digital

Digital (36, 128, 65)-net over F₄, using

t-expansion [i] based on digital (33, 128, 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, 128, 180)-Net over F₄ — Upper bound on s (digital)

There is no digital (36, 128, 181)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²⁸, 181, F₄, 92) (dual of [181, 53, 93]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹²⁷, 147, S₄, 92), but
    - the linear programming bound shows that MÂ â‰¥ 1349 953590 687270 246179 045330 695244 563410 842122 028765 631284 937718 372400 148416 386214 476266 864640 / 41158 876076 530201 > 4¹²⁷ [i]
  2. OA(4⁵³, 181, S₄, 34), but
    - discarding factors would yield OA(4⁵³, 172, S₄, 34), but
      - the linear programming bound shows that MÂ â‰¥ 123466 093718 521640 819257 910588 141484 804246 154591 951516 762688 716800 / 1516 591174 436793 940747 804532 901223 > 4⁵³ [i]

(36, 128, 248)-Net in Base 4 — Upper bound on s

There is no (36, 128, 249)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 135585 478127 236236 819734 300403 832969 132970 985972 888467 107197 676003 661941 154728 > 4¹²⁸ [i]