MinT - Best Known (39−21, 39, s)-Nets in Base 25

Best Known (39−21, 39, s)-Nets in Base 25

(39−21, 39, 132)-Net over F₂₅ — Constructive and digital

Digital (18, 39, 132)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 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]
2. digital (4, 25, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅ (see above)

(39−21, 39, 199)-Net over F₂₅ — Digital

Digital (18, 39, 199)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³⁹, 199, F₂₅, 21) (dual of [199, 160, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25³⁹, 208, F₂₅, 21) (dual of [208, 169, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 208Â | 25²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(39−21, 39, 38715)-Net in Base 25 — Upper bound on s

There is no (18, 39, 38716)-net in base 25, because

1 times m-reduction [i] would yield (18, 38, 38716)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 132374 906910 568428 120790 413582 198031 373822 275888 009153 > 25³⁸ [i]