MinT - Best Known (8, 18, s)-Nets in Base 256

Best Known (8, 18, s)-Nets in Base 256

(8, 18, 771)-Net over F₂₅₆ — Constructive and digital

Digital (8, 18, 771)-net over F₂₅₆, using

1 times m-reduction [i] based on 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, 18, 1285)-Net over F₂₅₆ — Digital

Digital (8, 18, 1285)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256¹⁸, 1285, F₂₅₆, 10) (dual of [1285, 1267, 11]-code), using
- the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,52,103,…,460}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(8, 18, 4774850)-Net in Base 256 — Upper bound on s

There is no (8, 18, 4774851)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 22 300761 552208 588660 627002 855344 375816 567776 > 256¹⁸ [i]