MinT - Best Known (62−23, 62, s)-Nets in Base 256

Best Known (62−23, 62, s)-Nets in Base 256

(62−23, 62, 6472)-Net over F₂₅₆ — Constructive and digital

Digital (39, 62, 6472)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 17, 515)-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 (1, 12, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (22, 45, 5957)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴⁵, 5957, F₂₅₆, 23, 23) (dual of [(5957, 23), 136966, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(256⁴⁵, 65528, F₂₅₆, 23) (dual of [65528, 65483, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁴⁵, 65536, F₂₅₆, 23) (dual of [65536, 65491, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(62−23, 62, 217367)-Net over F₂₅₆ — Digital

Digital (39, 62, 217367)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−23, 62, large)-Net in Base 256 — Upper bound on s

There is no (39, 62, large)-net in base 256, because

21 times m-reduction [i] would yield (39, 41, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]