MinT - Best Known (116−82, 116, s)-Nets in Base 4

Best Known (116−82, 116, s)-Nets in Base 4

(116−82, 116, 56)-Net over F₄ — Constructive and digital

Digital (34, 116, 56)-net over F₄, using

t-expansion [i] based on digital (33, 116, 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]

(116−82, 116, 65)-Net over F₄ — Digital

Digital (34, 116, 65)-net over F₄, using

t-expansion [i] based on digital (33, 116, 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]

(116−82, 116, 208)-Net over F₄ — Upper bound on s (digital)

There is no digital (34, 116, 209)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹¹⁶, 209, F₄, 82) (dual of [209, 93, 83]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹¹⁵, 144, S₄, 82), but
    - the linear programming bound shows that MÂ â‰¥ 1 809384 318048 793962 648861 468486 632853 295185 954954 708490 452357 715816 889075 089941 955770 056704 / 950 296664 568803 286553 > 4¹¹⁵ [i]
  2. OA(4⁹³, 209, S₄, 65), but
    - discarding factors would yield OA(4⁹³, 148, S₄, 65), but
      - the linear programming bound shows that MÂ â‰¥ 31410 303083 703049 377310 817051 099884 117194 259592 068916 365914 768905 433278 533937 431277 207276 796100 647129 523955 480347 541504 / 271 727597 647736 395513 057790 697511 115852 689190 837400 656430 005393 > 4⁹³ [i]

(116−82, 116, 239)-Net in Base 4 — Upper bound on s

There is no (34, 116, 240)-net in base 4, because

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]