MinT - Best Known (138−119, 138, s)-Nets in Base 5

Best Known (138−119, 138, s)-Nets in Base 5

(138−119, 138, 43)-Net over F₅ — Constructive and digital

Digital (19, 138, 43)-net over F₅, using

t-expansion [i] based on digital (18, 138, 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]

(138−119, 138, 45)-Net over F₅ — Digital

Digital (19, 138, 45)-net over F₅, using

net from sequence [i] based on digital (19, 44)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 19 and N(F) ≥ 45, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(138−119, 138, 101)-Net in Base 5 — Upper bound on s

There is no (19, 138, 102)-net in base 5, because

48 times m-reduction [i] would yield (19, 90, 102)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁹⁰, 102, S₅, 71), but
  - the linear programming bound shows that MÂ â‰¥ 18710 518494 394046 405477 542802 803236 909880 979510 489851 236343 383789 062500 / 22 710051 > 5⁹⁰ [i]