MinT - Best Known (68, 79, s)-Nets in Base 8

Best Known (68, 79, s)-Nets in Base 8

(68, 79, 1677734)-Net over F₈ — Constructive and digital

Digital (68, 79, 1677734)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [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]

(68, 79, large)-Net over F₈ — Digital

Digital (68, 79, large)-net over F₈, using

8⁶ 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]

(68, 79, large)-Net in Base 8 — Upper bound on s

There is no (68, 79, large)-net in base 8, because

9 times m-reduction [i] would yield (68, 70, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]