MinT - Best Known (136−109, 136, s)-Nets in Base 5

Best Known (136−109, 136, s)-Nets in Base 5

(136−109, 136, 51)-Net over F₅ — Constructive and digital

Digital (27, 136, 51)-net over F₅, using

t-expansion [i] based on digital (22, 136, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(136−109, 136, 55)-Net over F₅ — Digital

Digital (27, 136, 55)-net over F₅, using

t-expansion [i] based on digital (23, 136, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(136−109, 136, 137)-Net in Base 5 — Upper bound on s

There is no (27, 136, 138)-net in base 5, because

14 times m-reduction [i] would yield (27, 122, 138)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹²², 138, S₅, 95), but
  - the linear programming bound shows that MÂ â‰¥ 4298 350104 752369 269413 501600 679384 146953 677458 490777 611613 102571 208600 011232 192628 085613 250732 421875 / 221 979963 162672 > 5¹²² [i]