MinT - Best Known (112−15, 112, s)-Nets in Base 8

Best Known (112−15, 112, s)-Nets in Base 8

(112−15, 112, 1198380)-Net over F₈ — Constructive and digital

Digital (97, 112, 1198380)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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 (90, 105, 1198371)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁰⁵, 1198371, F₈, 15, 15) (dual of [(1198371, 15), 17975460, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(8¹⁰⁵, 8388598, F₈, 15) (dual of [8388598, 8388493, 16]-code), using
      - discarding factors / shortening the dual code 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]

(112−15, 112, large)-Net over F₈ — Digital

Digital (97, 112, large)-net over F₈, using

t-expansion [i] based on digital (96, 112, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹², large, F₈, 16) (dual of [large, large−112, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(112−15, 112, large)-Net in Base 8 — Upper bound on s

There is no (97, 112, large)-net in base 8, because

13 times m-reduction [i] would yield (97, 99, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]