MinT - Best Known (50−19, 50, s)-Nets in Base 256

Best Known (50−19, 50, s)-Nets in Base 256

(50−19, 50, 7795)-Net over F₂₅₆ — Constructive and digital

Digital (31, 50, 7795)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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, 9, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
2. digital (18, 37, 7281)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256³⁷, 7281, F₂₅₆, 19, 19) (dual of [(7281, 19), 138302, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(256³⁷, 65530, F₂₅₆, 19) (dual of [65530, 65493, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(256³⁷, 65536, F₂₅₆, 19) (dual of [65536, 65499, 20]-code), using
        an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(50−19, 50, 144935)-Net over F₂₅₆ — Digital

Digital (31, 50, 144935)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−19, 50, large)-Net in Base 256 — Upper bound on s

There is no (31, 50, large)-net in base 256, because

17 times m-reduction [i] would yield (31, 33, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]