MinT - Best Known (85, 85+173, s)-Nets in Base 4

Best Known (85, 85+173, s)-Nets in Base 4

(85, 85+173, 104)-Net over F₄ — Constructive and digital

Digital (85, 258, 104)-net over F₄, using

t-expansion [i] based on digital (73, 258, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(85, 85+173, 129)-Net over F₄ — Digital

Digital (85, 258, 129)-net over F₄, using

t-expansion [i] based on digital (81, 258, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(85, 85+173, 620)-Net in Base 4 — Upper bound on s

There is no (85, 258, 621)-net in base 4, because

1 times m-reduction [i] would yield (85, 257, 621)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 57884 125909 423238 592339 504942 771264 977727 147277 684262 811275 950171 693429 546036 448632 892623 134156 226066 940912 212578 030866 727008 528692 491495 933835 185757 055224 > 4²⁵⁷ [i]