MinT - Best Known (140−29, 140, s)-Nets in Base 8

Best Known (140−29, 140, s)-Nets in Base 8

(140−29, 140, 2349)-Net over F₈ — Constructive and digital

Digital (111, 140, 2349)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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 (97, 126, 2340)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²⁶, 2340, F₈, 29, 29) (dual of [(2340, 29), 67734, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(8¹²⁶, 32761, F₈, 29) (dual of [32761, 32635, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²⁶, 32768, F₈, 29) (dual of [32768, 32642, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(140−29, 140, 52902)-Net over F₈ — Digital

Digital (111, 140, 52902)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−29, 140, large)-Net in Base 8 — Upper bound on s

There is no (111, 140, large)-net in base 8, because

27 times m-reduction [i] would yield (111, 113, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]