MinT - Best Known (112, 112+20, s)-Nets in Base 8

Best Known (112, 112+20, s)-Nets in Base 8

(112, 112+20, 209732)-Net over F₈ — Constructive and digital

Digital (112, 132, 209732)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 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 (100, 120, 209715)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²⁰, 209715, F₈, 20, 20) (dual of [(209715, 20), 4194180, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(8¹²⁰, 2097150, F₈, 20) (dual of [2097150, 2097030, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²⁰, 2097152, F₈, 20) (dual of [2097152, 2097032, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(112, 112+20, 2129209)-Net over F₈ — Digital

Digital (112, 132, 2129209)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112, 112+20, large)-Net in Base 8 — Upper bound on s

There is no (112, 132, large)-net in base 8, because

18 times m-reduction [i] would yield (112, 114, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]