MinT - Best Known (70−14, 70, s)-Nets in Base 8

Best Known (70−14, 70, s)-Nets in Base 8

(70−14, 70, 4698)-Net over F₈ — Constructive and digital

Digital (56, 70, 4698)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 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 (47, 61, 4681)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶¹, 4681, F₈, 14, 14) (dual of [(4681, 14), 65473, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(8⁶¹, 32767, F₈, 14) (dual of [32767, 32706, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁶¹, 32768, F₈, 14) (dual of [32768, 32707, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(70−14, 70, 59040)-Net over F₈ — Digital

Digital (56, 70, 59040)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−14, 70, large)-Net in Base 8 — Upper bound on s

There is no (56, 70, large)-net in base 8, because

12 times m-reduction [i] would yield (56, 58, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]