MinT - Best Known (166−22, 166, s)-Nets in Base 8

Best Known (166−22, 166, s)-Nets in Base 8

(166−22, 166, 762617)-Net over F₈ — Constructive and digital

Digital (144, 166, 762617)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (131, 153, 762600)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁵³, 762600, F₈, 22, 22) (dual of [(762600, 22), 16777047, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(8¹⁵³, 8388600, F₈, 22) (dual of [8388600, 8388447, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(166−22, 166, large)-Net over F₈ — Digital

Digital (144, 166, large)-net over F₈, using

2 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(166−22, 166, large)-Net in Base 8 — Upper bound on s

There is no (144, 166, large)-net in base 8, because

20 times m-reduction [i] would yield (144, 146, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]