MinT - Best Known (99−3, 99, s)-Nets in Base 8

Best Known (99−3, 99, s)-Nets in Base 8

(99−3, 99, large)-Net over F₈ — Constructive and digital

Digital (96, 99, large)-net over F₈, using

8⁴ times duplication [i] based on digital (92, 95, large)-net over F₈, using
- t-expansion [i] based on digital (86, 95, large)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (21, 25, 4194301)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8²⁵, 4194301, F₈, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(8²⁵, 4194301, F₈, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(8²⁵, 8388602, F₈, 4) (dual of [8388602, 8388577, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(8²⁵, large, F₈, 4) (dual of [large, large−25, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    2. digital (61, 70, 4194309)-net over F₈, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 4, 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]
        digital (57, 66, 4194300)-net over F₈, using
        net defined by OOA [i] based on linear OOA(8⁶⁶, 4194300, F₈, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁶⁶, 8388601, F₈, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8⁶⁶, 8388602, F₈, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
        trace code [i] based on linear OOA(64³³, 4194301, F₆₄, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³³, 8388602, F₆₄, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(99−3, 99, large)-Net in Base 8 — Upper bound on s

There is no (96, 99, large)-net in base 8, because

1 times m-reduction [i] would yield (96, 98, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (99−3, 99, s)-Nets in Base 8

(99−3, 99, large)-Net over F8 — Constructive and digital

(99−3, 99, large)-Net in Base 8 — Upper bound on s

(99−3, 99, large)-Net over F₈ — Constructive and digital