MinT - Best Known (137−109, 137, s)-Nets in Base 5

Best Known (137−109, 137, s)-Nets in Base 5

(137−109, 137, 51)-Net over F₅ — Constructive and digital

Digital (28, 137, 51)-net over F₅, using

t-expansion [i] based on digital (22, 137, 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]

(137−109, 137, 55)-Net over F₅ — Digital

Digital (28, 137, 55)-net over F₅, using

t-expansion [i] based on digital (23, 137, 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]

(137−109, 137, 142)-Net in Base 5 — Upper bound on s

There is no (28, 137, 143)-net in base 5, because

10 times m-reduction [i] would yield (28, 127, 143)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹²⁷, 143, S₅, 99), but
  - the linear programming bound shows that MÂ â‰¥ 16790 877570 883690 780699 267730 530691 899698 390565 322722 646512 433597 454361 233758 390881 121158 599853 515625 / 225753 948288 > 5¹²⁷ [i]