MinT - Best Known (34, 34+20, s)-Nets in Base 256

Best Known (34, 34+20, s)-Nets in Base 256

(34, 34+20, 7067)-Net over F₂₅₆ — Constructive and digital

Digital (34, 54, 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 (19, 39, 6553)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256³⁹, 6553, F₂₅₆, 20, 20) (dual of [(6553, 20), 131021, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(256³⁹, 65530, F₂₅₆, 20) (dual of [65530, 65491, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(256³⁹, 65536, F₂₅₆, 20) (dual of [65536, 65497, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(34, 34+20, 217353)-Net over F₂₅₆ — Digital

Digital (34, 54, 217353)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+20, large)-Net in Base 256 — Upper bound on s

There is no (34, 54, large)-net in base 256, because

18 times m-reduction [i] would yield (34, 36, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]