MinT - Best Known (139, 163, s)-Nets in Base 8

Best Known (139, 163, s)-Nets in Base 8

(139, 163, 174787)-Net over F₈ — Constructive and digital

Digital (139, 163, 174787)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 16, 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 (123, 147, 174762)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁷, 174762, F₈, 24, 24) (dual of [(174762, 24), 4194141, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(8¹⁴⁷, 2097144, F₈, 24) (dual of [2097144, 2096997, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴⁷, 2097151, F₈, 24) (dual of [2097151, 2097004, 25]-code), using
        1 times truncation [i] based on linear OA(8¹⁴⁸, 2097152, F₈, 25) (dual of [2097152, 2097004, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(139, 163, 3384692)-Net over F₈ — Digital

Digital (139, 163, 3384692)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 163, large)-Net in Base 8 — Upper bound on s

There is no (139, 163, large)-net in base 8, because

22 times m-reduction [i] would yield (139, 141, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]