MinT - Best Known (75−16, 75, s)-Nets in Base 25

Best Known (75−16, 75, s)-Nets in Base 25

(75−16, 75, 48906)-Net over F₂₅ — Constructive and digital

Digital (59, 75, 48906)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 14, 78)-net over F₂₅, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 26)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
    2. digital (0, 4, 26)-net over F₂₅, using
      - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
        Niederreiter sequence [i]
    3. digital (0, 8, 26)-net over F₂₅, using
      - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
2. digital (45, 61, 48828)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶¹, 48828, F₂₅, 16, 16) (dual of [(48828, 16), 781187, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(25⁶¹, 390624, F₂₅, 16) (dual of [390624, 390563, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶¹, 390625, F₂₅, 16) (dual of [390625, 390564, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(75−16, 75, 2613706)-Net over F₂₅ — Digital

Digital (59, 75, 2613706)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−16, 75, large)-Net in Base 25 — Upper bound on s

There is no (59, 75, large)-net in base 25, because

14 times m-reduction [i] would yield (59, 61, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]