MinT - Best Known (8, 19, s)-Nets in Base 256

Best Known (8, 19, s)-Nets in Base 256

(8, 19, 771)-Net over F₂₅₆ — Constructive and digital

Digital (8, 19, 771)-net over F₂₅₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 3, 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, 5, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
3. digital (0, 11, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(8, 19, 1062)-Net over F₂₅₆ — Digital

Digital (8, 19, 1062)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256¹⁹, 1062, F₂₅₆, 11) (dual of [1062, 1043, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(256¹⁹, 1285, F₂₅₆, 11) (dual of [1285, 1266, 12]-code), using
  - the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,155,309,…,256}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(8, 19, 4774850)-Net in Base 256 — Upper bound on s

There is no (8, 19, 4774851)-net in base 256, because

1 times m-reduction [i] would yield (8, 18, 4774851)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 22 300761 552208 588660 627002 855344 375816 567776 > 256¹⁸ [i]