MinT - Best Known (52, 52+19, s)-Nets in Base 25

Best Known (52, 52+19, s)-Nets in Base 25

(52, 52+19, 1814)-Net over F₂₅ — Constructive and digital

Digital (52, 71, 1814)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 16, 78)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (3, 12, 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]
2. digital (36, 55, 1736)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁵⁵, 1736, F₂₅, 19, 19) (dual of [(1736, 19), 32929, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(25⁵⁵, 15625, F₂₅, 19) (dual of [15625, 15570, 20]-code), using
      - an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(52, 52+19, 102814)-Net over F₂₅ — Digital

Digital (52, 71, 102814)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+19, large)-Net in Base 25 — Upper bound on s

There is no (52, 71, large)-net in base 25, because

17 times m-reduction [i] would yield (52, 54, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]