MinT - Best Known (166−29, 166, s)-Nets in Base 8

Best Known (166−29, 166, s)-Nets in Base 8

(166−29, 166, 18738)-Net over F₈ — Constructive and digital

Digital (137, 166, 18738)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 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 (122, 151, 18724)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁵¹, 18724, F₈, 29, 29) (dual of [(18724, 29), 542845, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(8¹⁵¹, 262137, F₈, 29) (dual of [262137, 261986, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁵¹, 262144, F₈, 29) (dual of [262144, 261993, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(166−29, 166, 364718)-Net over F₈ — Digital

Digital (137, 166, 364718)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−29, 166, large)-Net in Base 8 — Upper bound on s

There is no (137, 166, large)-net in base 8, because

27 times m-reduction [i] would yield (137, 139, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]