MinT - Best Known (56−21, 56, s)-Nets in Base 256

Best Known (56−21, 56, s)-Nets in Base 256

(56−21, 56, 7067)-Net over F₂₅₆ — Constructive and digital

Digital (35, 56, 7067)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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, 10, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
2. digital (20, 41, 6553)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴¹, 6553, F₂₅₆, 21, 21) (dual of [(6553, 21), 137572, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(256⁴¹, 65531, F₂₅₆, 21) (dual of [65531, 65490, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁴¹, 65536, F₂₅₆, 21) (dual of [65536, 65495, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(56−21, 56, 180245)-Net over F₂₅₆ — Digital

Digital (35, 56, 180245)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−21, 56, large)-Net in Base 256 — Upper bound on s

There is no (35, 56, large)-net in base 256, because

19 times m-reduction [i] would yield (35, 37, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]