MinT - Best Known (45−25, 45, s)-Nets in Base 256

Best Known (45−25, 45, s)-Nets in Base 256

(45−25, 45, 771)-Net over F₂₅₆ — Constructive and digital

Digital (20, 45, 771)-net over F₂₅₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 8, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 12, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
3. digital (0, 25, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(45−25, 45, 1285)-Net over F₂₅₆ — Digital

Digital (20, 45, 1285)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁴⁵, 1285, F₂₅₆, 25) (dual of [1285, 1240, 26]-code), using
- the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,78,155,…,564}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(45−25, 45, large)-Net in Base 256 — Upper bound on s

There is no (20, 45, large)-net in base 256, because

23 times m-reduction [i] would yield (20, 22, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]