MinT - Best Known (87−12, 87, s)-Nets in Base 8

Best Known (87−12, 87, s)-Nets in Base 8

(87−12, 87, 1398109)-Net over F₈ — Constructive and digital

Digital (75, 87, 1398109)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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 (69, 81, 1398100)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁸¹, 1398100, F₈, 12, 12) (dual of [(1398100, 12), 16777119, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(8⁸¹, 8388600, F₈, 12) (dual of [8388600, 8388519, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁸¹, large, F₈, 12) (dual of [large, large−81, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(87−12, 87, large)-Net over F₈ — Digital

Digital (75, 87, large)-net over F₈, using

8⁶ times duplication [i] based on digital (69, 81, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸¹, large, F₈, 12) (dual of [large, large−81, 13]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(87−12, 87, large)-Net in Base 8 — Upper bound on s

There is no (75, 87, large)-net in base 8, because

10 times m-reduction [i] would yield (75, 77, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]