MinT - Best Known (111−85, 111, s)-Nets in Base 5

Best Known (111−85, 111, s)-Nets in Base 5

(111−85, 111, 51)-Net over F₅ — Constructive and digital

Digital (26, 111, 51)-net over F₅, using

t-expansion [i] based on digital (22, 111, 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]

(111−85, 111, 55)-Net over F₅ — Digital

Digital (26, 111, 55)-net over F₅, using

t-expansion [i] based on digital (23, 111, 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]

(111−85, 111, 142)-Net in Base 5 — Upper bound on s

There is no (26, 111, 143)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5¹¹¹, 143, S₅, 85), but
- the linear programming bound shows that MÂ â‰¥ 220 445839 236062 243251 966630 886846 194373 438661 017264 989094 781670 909972 692840 028685 168363 153934 478759 765625 / 378 761134 914432 479453 277033 > 5¹¹¹ [i]