MinT - Best Known (89−8, 89, s)-Nets in Base 8

Best Known (89−8, 89, s)-Nets in Base 8

(89−8, 89, large)-Net over F₈ — Constructive and digital

Digital (81, 89, large)-net over F₈, using

8² times duplication [i] based on digital (79, 87, 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 (54, 62, 4194309)-net over F₈, using
    - (u,Â u+v)-construction [i] based on
      1. 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]
      2. digital (50, 58, 4194300)-net over F₈, using
        net defined by OOA [i] based on linear OOA(8⁵⁸, 4194300, F₈, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁵⁸, 8388601, F₈, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8⁵⁸, 8388602, F₈, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
        trace code [i] based on linear OOA(64²⁹, 4194301, F₆₄, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²⁹, 8388602, F₆₄, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(89−8, 89, large)-Net in Base 8 — Upper bound on s

There is no (81, 89, large)-net in base 8, because

6 times m-reduction [i] would yield (81, 83, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (89−8, 89, s)-Nets in Base 8

(89−8, 89, large)-Net over F8 — Constructive and digital

(89−8, 89, large)-Net in Base 8 — Upper bound on s

(89−8, 89, large)-Net over F₈ — Constructive and digital