MinT - Best Known (117−90, 117, s)-Nets in Base 5

Best Known (117−90, 117, s)-Nets in Base 5

(117−90, 117, 51)-Net over F₅ — Constructive and digital

Digital (27, 117, 51)-net over F₅, using

t-expansion [i] based on digital (22, 117, 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]

(117−90, 117, 55)-Net over F₅ — Digital

Digital (27, 117, 55)-net over F₅, using

t-expansion [i] based on digital (23, 117, 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]

(117−90, 117, 142)-Net in Base 5 — Upper bound on s

There is no (27, 117, 143)-net in base 5, because

1 times m-reduction [i] would yield (27, 116, 143)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5¹¹⁶, 143, S₅, 89), but
  - the linear programming bound shows that MÂ â‰¥ 2738 237588 593697 454668 762256 522048 887080 570758 264064 445890 203342 535984 543104 632393 806241 452693 939208 984375 / 2 228541 507840 976583 547528 > 5¹¹⁶ [i]