MinT - Best Known (52−14, 52, s)-Nets in Base 25

Best Known (52−14, 52, s)-Nets in Base 25

(52−14, 52, 2310)-Net over F₂₅ — Constructive and digital

Digital (38, 52, 2310)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 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, 3, 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, 7, 26)-net over F₂₅, using
      - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
2. digital (26, 40, 2232)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴⁰, 2232, F₂₅, 14, 14) (dual of [(2232, 14), 31208, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(25⁴⁰, 15624, F₂₅, 14) (dual of [15624, 15584, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁴⁰, 15625, F₂₅, 14) (dual of [15625, 15585, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(52−14, 52, 92254)-Net over F₂₅ — Digital

Digital (38, 52, 92254)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−14, 52, large)-Net in Base 25 — Upper bound on s

There is no (38, 52, large)-net in base 25, because

12 times m-reduction [i] would yield (38, 40, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]