MinT - Best Known (56, 78, s)-Nets in Base 25

Best Known (56, 78, s)-Nets in Base 25

(56, 78, 1472)-Net over F₂₅ — Constructive and digital

Digital (56, 78, 1472)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 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 (42, 64, 1420)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶⁴, 1420, F₂₅, 22, 22) (dual of [(1420, 22), 31176, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(25⁶⁴, 15620, F₂₅, 22) (dual of [15620, 15556, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶⁴, 15625, F₂₅, 22) (dual of [15625, 15561, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(56, 78, 56326)-Net over F₂₅ — Digital

Digital (56, 78, 56326)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 78, large)-Net in Base 25 — Upper bound on s

There is no (56, 78, large)-net in base 25, because

20 times m-reduction [i] would yield (56, 58, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]