MinT - Best Known (68−25, 68, s)-Nets in Base 256

Best Known (68−25, 68, s)-Nets in Base 256

(68−25, 68, 5976)-Net over F₂₅₆ — Constructive and digital

Digital (43, 68, 5976)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (7, 19, 515)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 6, 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 (1, 13, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (24, 49, 5461)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴⁹, 5461, F₂₅₆, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(256⁴⁹, 65533, F₂₅₆, 25) (dual of [65533, 65484, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁴⁹, 65536, F₂₅₆, 25) (dual of [65536, 65487, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(68−25, 68, 255970)-Net over F₂₅₆ — Digital

Digital (43, 68, 255970)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−25, 68, large)-Net in Base 256 — Upper bound on s

There is no (43, 68, large)-net in base 256, because

23 times m-reduction [i] would yield (43, 45, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]