MinT - Best Known (71−10, 71, s)-Nets in Base 8

Best Known (71−10, 71, s)-Nets in Base 8

(71−10, 71, 1677734)-Net over F₈ — Constructive and digital

Digital (61, 71, 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 (55, 65, 1677720)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶⁵, 1677720, F₈, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(8⁶⁵, 8388600, F₈, 10) (dual of [8388600, 8388535, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁶⁵, large, F₈, 10) (dual of [large, large−65, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(71−10, 71, large)-Net over F₈ — Digital

Digital (61, 71, large)-net over F₈, using

8⁶ times duplication [i] based on digital (55, 65, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁵, large, F₈, 10) (dual of [large, large−65, 11]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(71−10, 71, large)-Net in Base 8 — Upper bound on s

There is no (61, 71, large)-net in base 8, because

8 times m-reduction [i] would yield (61, 63, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]