MinT - Best Known (43−12, 43, s)-Nets in Base 25

Best Known (43−12, 43, s)-Nets in Base 25

(43−12, 43, 2656)-Net over F₂₅ — Constructive and digital

Digital (31, 43, 2656)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 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 (22, 34, 2604)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25³⁴, 2604, F₂₅, 12, 12) (dual of [(2604, 12), 31214, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(25³⁴, 15624, F₂₅, 12) (dual of [15624, 15590, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(25³⁴, 15625, F₂₅, 12) (dual of [15625, 15591, 13]-code), using
        an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(43−12, 43, 59637)-Net over F₂₅ — Digital

Digital (31, 43, 59637)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(43−12, 43, large)-Net in Base 25 — Upper bound on s

There is no (31, 43, large)-net in base 25, because

10 times m-reduction [i] would yield (31, 33, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]