MinT - Best Known (23, 68, s)-Nets in Base 4

Best Known (23, 68, s)-Nets in Base 4

(23, 68, 34)-Net over F₄ — Constructive and digital

Digital (23, 68, 34)-net over F₄, using

t-expansion [i] based on digital (21, 68, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(23, 68, 45)-Net over F₄ — Digital

Digital (23, 68, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(23, 68, 179)-Net in Base 4 — Upper bound on s

There is no (23, 68, 180)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁸, 180, S₄, 45), but
- the linear programming bound shows that MÂ â‰¥ 328494 790735 001198 305199 602445 532392 471737 863161 499929 581656 031112 774235 621273 901469 693549 435944 960000 000000 / 3 747617 146367 023709 191641 153229 380578 240235 288823 915002 005290 401803 > 4⁶⁸ [i]