MinT - Best Known (10, 10+13, s)-Nets in Base 256

Best Known (10, 10+13, s)-Nets in Base 256

(10, 10+13, 771)-Net over F₂₅₆ — Constructive and digital

Digital (10, 23, 771)-net over F₂₅₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 4, 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, 6, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
3. digital (0, 13, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(10, 10+13, 1257)-Net over F₂₅₆ — Digital

Digital (10, 23, 1257)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256²³, 1257, F₂₅₆, 13) (dual of [1257, 1234, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(256²³, 1285, F₂₅₆, 13) (dual of [1285, 1262, 14]-code), using
  - the BCH-code C(I) with length 1285Â | 256²âˆ’1, defining interval IÂ =Â {1,155,309,…,564}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(10, 10+13, 7941361)-Net in Base 256 — Upper bound on s

There is no (10, 23, 7941362)-net in base 256, because

1 times m-reduction [i] would yield (10, 22, 7941362)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 95781 001688 816224 435782 396924 005862 493769 803809 813736 > 256²² [i]