MinT - Best Known (27, 121, s)-Nets in Base 5

Best Known (27, 121, s)-Nets in Base 5

(27, 121, 51)-Net over F₅ — Constructive and digital

Digital (27, 121, 51)-net over F₅, using

t-expansion [i] based on digital (22, 121, 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]

(27, 121, 55)-Net over F₅ — Digital

Digital (27, 121, 55)-net over F₅, using

t-expansion [i] based on digital (23, 121, 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]

(27, 121, 138)-Net in Base 5 — Upper bound on s

There is no (27, 121, 139)-net in base 5, because

1 times m-reduction [i] would yield (27, 120, 139)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹²⁰, 139, S₅, 93), but
  - the linear programming bound shows that MÂ â‰¥ 27858 012410 071478 273208 995833 145273 886678 323269 030897 943996 684384 959650 060409 330762 922763 824462 890625 / 33773 631432 131241 > 5¹²⁰ [i]