MinT - Best Known (67, 78, s)-Nets in Base 8

Best Known (67, 78, s)-Nets in Base 8

(67, 78, 1677729)-Net over F₈ — Constructive and digital

Digital (67, 78, 1677729)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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 (62, 73, 1677720)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁷³, 1677720, F₈, 11, 11) (dual of [(1677720, 11), 18454847, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(8⁷³, 8388601, F₈, 11) (dual of [8388601, 8388528, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁷³, large, F₈, 11) (dual of [large, large−73, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(67, 78, large)-Net over F₈ — Digital

Digital (67, 78, large)-net over F₈, using

8⁵ times duplication [i] based on digital (62, 73, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷³, large, F₈, 11) (dual of [large, large−73, 12]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(67, 78, large)-Net in Base 8 — Upper bound on s

There is no (67, 78, large)-net in base 8, because

9 times m-reduction [i] would yield (67, 69, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]