MinT - Best Known (16, 16+9, s)-Nets in Base 5

Best Known (16, 16+9, s)-Nets in Base 5

(16, 16+9, 104)-Net over F₅ — Constructive and digital

Digital (16, 25, 104)-net over F₅, using

1 times m-reduction [i] based on digital (16, 26, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 13, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]

(16, 16+9, 159)-Net over F₅ — Digital

Digital (16, 25, 159)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²⁵, 159, F₅, 9) (dual of [159, 134, 10]-code), using
- 29 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 14 times 0) [i] based on linear OA(5²⁰, 125, F₅, 9) (dual of [125, 105, 10]-code), using
  - a â€œGraXâ€ code from Grasslâ€™s database [i]

(16, 16+9, 8643)-Net in Base 5 — Upper bound on s

There is no (16, 25, 8644)-net in base 5, because

1 times m-reduction [i] would yield (16, 24, 8644)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 59619 736020 433665 > 5²⁴ [i]