MinT - Best Known (13, 30, s)-Nets in Base 256

Best Known (13, 30, s)-Nets in Base 256

(13, 30, 771)-Net over F₂₅₆ — Constructive and digital

Digital (13, 30, 771)-net over F₂₅₆, using

generalized (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, 8, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
3. digital (0, 17, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(13, 30, 1134)-Net over F₂₅₆ — Digital

Digital (13, 30, 1134)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256³⁰, 1134, F₂₅₆, 17) (dual of [1134, 1104, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(256³⁰, 1285, F₂₅₆, 17) (dual of [1285, 1255, 18]-code), using
  - the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,18,35,…,273}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(13, 30, 7925370)-Net in Base 256 — Upper bound on s

There is no (13, 30, 7925371)-net in base 256, because

1 times m-reduction [i] would yield (13, 29, 7925371)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 6901 750883 554126 532838 170881 991747 788913 019877 362480 132359 379259 796791 > 256²⁹ [i]