MinT - Best Known (99−21, 99, s)-Nets in Base 25

Best Known (99−21, 99, s)-Nets in Base 25

(99−21, 99, 39140)-Net over F₂₅ — Constructive and digital

Digital (78, 99, 39140)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 78)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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, 13, 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 (60, 81, 39062)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁸¹, 39062, F₂₅, 21, 21) (dual of [(39062, 21), 820221, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(25⁸¹, 390621, F₂₅, 21) (dual of [390621, 390540, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁸¹, 390625, F₂₅, 21) (dual of [390625, 390544, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(99−21, 99, 2876733)-Net over F₂₅ — Digital

Digital (78, 99, 2876733)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−21, 99, large)-Net in Base 25 — Upper bound on s

There is no (78, 99, large)-net in base 25, because

19 times m-reduction [i] would yield (78, 80, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (99−21, 99, s)-Nets in Base 25

(99−21, 99, 39140)-Net over F25 — Constructive and digital

(99−21, 99, 2876733)-Net over F25 — Digital

(99−21, 99, large)-Net in Base 25 — Upper bound on s

(99−21, 99, 39140)-Net over F₂₅ — Constructive and digital

(99−21, 99, 2876733)-Net over F₂₅ — Digital