MinT - Best Known (171−35, 171, s)-Nets in Base 8

Best Known (171−35, 171, s)-Nets in Base 8

(171−35, 171, 1951)-Net over F₈ — Constructive and digital

Digital (136, 171, 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 (116, 151, 1927)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁵¹, 1927, F₈, 35, 35) (dual of [(1927, 35), 67294, 36]-NRT-code), using
    - OOA 17-folding and stacking with additional row [i] based on linear OA(8¹⁵¹, 32760, F₈, 35) (dual of [32760, 32609, 36]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁵¹, 32768, F₈, 35) (dual of [32768, 32617, 36]-code), using
        an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(171−35, 171, 67375)-Net over F₈ — Digital

Digital (136, 171, 67375)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−35, 171, large)-Net in Base 8 — Upper bound on s

There is no (136, 171, large)-net in base 8, because

33 times m-reduction [i] would yield (136, 138, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]