MinT - Best Known (105−14, 105, s)-Nets in Base 8

Best Known (105−14, 105, s)-Nets in Base 8

(105−14, 105, 1198385)-Net over F₈ — Constructive and digital

Digital (91, 105, 1198385)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 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 (83, 97, 1198371)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁷, 1198371, F₈, 14, 14) (dual of [(1198371, 14), 16777097, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(8⁹⁷, 8388597, F₈, 14) (dual of [8388597, 8388500, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁹⁷, large, F₈, 14) (dual of [large, large−97, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(105−14, 105, large)-Net over F₈ — Digital

Digital (91, 105, large)-net over F₈, using

t-expansion [i] based on digital (90, 105, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁵, large, F₈, 15) (dual of [large, large−105, 16]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(105−14, 105, large)-Net in Base 8 — Upper bound on s

There is no (91, 105, large)-net in base 8, because

12 times m-reduction [i] would yield (91, 93, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]