MinT - Best Known (89−69, 89, s)-Nets in Base 5

Best Known (89−69, 89, s)-Nets in Base 5

(89−69, 89, 43)-Net over F₅ — Constructive and digital

Digital (20, 89, 43)-net over F₅, using

t-expansion [i] based on digital (18, 89, 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]

(89−69, 89, 45)-Net over F₅ — Digital

Digital (20, 89, 45)-net over F₅, using

t-expansion [i] based on digital (19, 89, 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]

(89−69, 89, 111)-Net in Base 5 — Upper bound on s

There is no (20, 89, 112)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁸⁹, 112, S₅, 69), but
- the linear programming bound shows that MÂ â‰¥ 80482 634717 824845 779643 893028 051384 284926 594858 688986 278139 054775 238037 109375 / 378 614401 736571 > 5⁸⁹ [i]