MinT - Best Known (28−9, 28, s)-Nets in Base 5

Best Known (28−9, 28, s)-Nets in Base 5

(28−9, 28, 132)-Net over F₅ — Constructive and digital

Digital (19, 28, 132)-net over F₅, using

2 times m-reduction [i] based on digital (19, 30, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 15, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(28−9, 28, 268)-Net over F₅ — Digital

Digital (19, 28, 268)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²⁸, 268, F₅, 9) (dual of [268, 240, 10]-code), using
- 135 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 14 times 0, 1, 24 times 0, 1, 34 times 0, 1, 45 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]

(28−9, 28, 28906)-Net in Base 5 — Upper bound on s

There is no (19, 28, 28907)-net in base 5, because

1 times m-reduction [i] would yield (19, 27, 28907)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 450596 434452 913265 > 5²⁷ [i]