MinT - Best Known (162−24, 162, s)-Nets in Base 8

Best Known (162−24, 162, s)-Nets in Base 8

(162−24, 162, 174786)-Net over F₈ — Constructive and digital

Digital (138, 162, 174786)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 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 (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]

(162−24, 162, 3092107)-Net over F₈ — Digital

Digital (138, 162, 3092107)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−24, 162, large)-Net in Base 8 — Upper bound on s

There is no (138, 162, large)-net in base 8, because

22 times m-reduction [i] would yield (138, 140, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]