MinT - Best Known (72−51, 72, s)-Nets in Base 5

Best Known (72−51, 72, s)-Nets in Base 5

(72−51, 72, 43)-Net over F₅ — Constructive and digital

Digital (21, 72, 43)-net over F₅, using

t-expansion [i] based on digital (18, 72, 43)-net over F₅, using
- net from sequence [i] based on digital (18, 42)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 17, N(F)Â =Â 42, and 1 place with degree 2 [i] based on function field F/F₅ with g(F) = 17 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(72−51, 72, 50)-Net over F₅ — Digital

Digital (21, 72, 50)-net over F₅, using

net from sequence [i] based on digital (21, 49)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 21 and N(F) ≥ 50, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(72−51, 72, 226)-Net in Base 5 — Upper bound on s

There is no (21, 72, 227)-net in base 5, because

3 times m-reduction [i] would yield (21, 69, 227)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁶⁹, 227, S₅, 48), but
  - the linear programming bound shows that MÂ â‰¥ 43120 034420 356247 799670 563974 686565 677107 191594 132870 542557 401781 346057 168107 588353 223036 392591 893672 943115 234375 / 25082 081198 310689 525487 009337 454523 517550 452334 855551 408964 686501 > 5⁶⁹ [i]