MinT - Best Known (69, 69+25, s)-Nets in Base 25

Best Known (69, 69+25, s)-Nets in Base 25

(69, 69+25, 1427)-Net over F₂₅ — Constructive and digital

Digital (69, 94, 1427)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 22, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]
2. digital (47, 72, 1301)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷², 1301, F₂₅, 25, 25) (dual of [(1301, 25), 32453, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(25⁷², 15613, F₂₅, 25) (dual of [15613, 15541, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁷², 15624, F₂₅, 25) (dual of [15624, 15552, 26]-code), using
        1 times truncation [i] based on linear OA(25⁷³, 15625, F₂₅, 26) (dual of [15625, 15552, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(69, 69+25, 122028)-Net over F₂₅ — Digital

Digital (69, 94, 122028)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+25, large)-Net in Base 25 — Upper bound on s

There is no (69, 94, large)-net in base 25, because

23 times m-reduction [i] would yield (69, 71, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]