MinT - Best Known (65, 65+164, s)-Nets in Base 4

Best Known (65, 65+164, s)-Nets in Base 4

(65, 65+164, 66)-Net over F₄ — Constructive and digital

Digital (65, 229, 66)-net over F₄, using

t-expansion [i] based on digital (49, 229, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(65, 65+164, 99)-Net over F₄ — Digital

Digital (65, 229, 99)-net over F₄, using

t-expansion [i] based on digital (61, 229, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(65, 65+164, 386)-Net over F₄ — Upper bound on s (digital)

There is no digital (65, 229, 387)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁹, 387, F₄, 164) (dual of [387, 158, 165]-code), but
- residual code [i] would yield OA(4⁶⁵, 222, S₄, 41), but
  - the linear programming bound shows that MÂ â‰¥ 32 342197 523814 651165 392954 045591 141352 749935 799714 032461 858391 219530 982071 151362 048000 / 23186 656803 327364 360991 616330 320246 466025 092007 > 4⁶⁵ [i]

(65, 65+164, 436)-Net in Base 4 — Upper bound on s

There is no (65, 229, 437)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 747501 651698 145615 683400 643757 719088 248991 136622 455012 003183 532071 666863 778279 001300 810739 184899 816556 943232 210849 340200 838072 585423 087100 > 4²²⁹ [i]