MinT - Best Known (69−14, 69, s)-Nets in Base 8

Best Known (69−14, 69, s)-Nets in Base 8

(69−14, 69, 4695)-Net over F₈ — Constructive and digital

Digital (55, 69, 4695)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [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]

(69−14, 69, 50314)-Net over F₈ — Digital

Digital (55, 69, 50314)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−14, 69, large)-Net in Base 8 — Upper bound on s

There is no (55, 69, large)-net in base 8, because

12 times m-reduction [i] would yield (55, 57, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]