MinT - Best Known (172−26, 172, s)-Nets in Base 8

Best Known (172−26, 172, s)-Nets in Base 8

(172−26, 172, 161344)-Net over F₈ — Constructive and digital

Digital (146, 172, 161344)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (129, 155, 161319)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁵⁵, 161319, F₈, 26, 26) (dual of [(161319, 26), 4194139, 27]-NRT-code), using
    - OA 13-folding and stacking [i] based on linear OA(8¹⁵⁵, 2097147, F₈, 26) (dual of [2097147, 2096992, 27]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁵⁵, 2097152, F₈, 26) (dual of [2097152, 2096997, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(172−26, 172, 2375687)-Net over F₈ — Digital

Digital (146, 172, 2375687)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−26, 172, large)-Net in Base 8 — Upper bound on s

There is no (146, 172, large)-net in base 8, because

24 times m-reduction [i] would yield (146, 148, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]