MinT - Best Known (166−34, 166, s)-Nets in Base 8

Best Known (166−34, 166, s)-Nets in Base 8

(166−34, 166, 1951)-Net over F₈ — Constructive and digital

Digital (132, 166, 1951)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (112, 146, 1927)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁶, 1927, F₈, 34, 34) (dual of [(1927, 34), 65372, 35]-NRT-code), using
    - OA 17-folding and stacking [i] based on linear OA(8¹⁴⁶, 32759, F₈, 34) (dual of [32759, 32613, 35]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴⁶, 32768, F₈, 34) (dual of [32768, 32622, 35]-code), using
        an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(166−34, 166, 65642)-Net over F₈ — Digital

Digital (132, 166, 65642)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−34, 166, large)-Net in Base 8 — Upper bound on s

There is no (132, 166, large)-net in base 8, because

32 times m-reduction [i] would yield (132, 134, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]