MinT - Best Known (65−15, 65, s)-Nets in Base 8

Best Known (65−15, 65, s)-Nets in Base 8

(65−15, 65, 1179)-Net over F₈ — Constructive and digital

Digital (50, 65, 1179)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (43, 58, 1170)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁸, 1170, F₈, 15, 15) (dual of [(1170, 15), 17492, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(8⁵⁸, 8191, F₈, 15) (dual of [8191, 8133, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁵⁸, 8194, F₈, 15) (dual of [8194, 8136, 16]-code), using
        trace code [i] based on linear OA(64²⁹, 4097, F₆₄, 15) (dual of [4097, 4068, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(65−15, 65, 13474)-Net over F₈ — Digital

Digital (50, 65, 13474)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−15, 65, large)-Net in Base 8 — Upper bound on s

There is no (50, 65, large)-net in base 8, because

13 times m-reduction [i] would yield (50, 52, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]