MinT - Best Known (59−22, 59, s)-Nets in Base 256

Best Known (59−22, 59, s)-Nets in Base 256

(59−22, 59, 6472)-Net over F₂₅₆ — Constructive and digital

Digital (37, 59, 6472)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 16, 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, 11, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
2. digital (21, 43, 5958)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴³, 5958, F₂₅₆, 22, 22) (dual of [(5958, 22), 131033, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(256⁴³, 65538, F₂₅₆, 22) (dual of [65538, 65495, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
        linear OA(256⁴³, 65536, F₂₅₆, 22) (dual of [65536, 65493, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        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]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(59−22, 59, 198602)-Net over F₂₅₆ — Digital

Digital (37, 59, 198602)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−22, 59, large)-Net in Base 256 — Upper bound on s

There is no (37, 59, large)-net in base 256, because

20 times m-reduction [i] would yield (37, 39, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (59−22, 59, s)-Nets in Base 256

(59−22, 59, 6472)-Net over F256 — Constructive and digital

(59−22, 59, 198602)-Net over F256 — Digital

(59−22, 59, large)-Net in Base 256 — Upper bound on s

(59−22, 59, 6472)-Net over F₂₅₆ — Constructive and digital

(59−22, 59, 198602)-Net over F₂₅₆ — Digital