MinT - Best Known (173−30, 173, s)-Nets in Base 8

Best Known (173−30, 173, s)-Nets in Base 8

(173−30, 173, 17490)-Net over F₈ — Constructive and digital

Digital (143, 173, 17490)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 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 (127, 157, 17476)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁵⁷, 17476, F₈, 30, 30) (dual of [(17476, 30), 524123, 31]-NRT-code), using
    - OA 15-folding and stacking [i] based on linear OA(8¹⁵⁷, 262140, F₈, 30) (dual of [262140, 261983, 31]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁵⁷, 262144, F₈, 30) (dual of [262144, 261987, 31]-code), using
        an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(173−30, 173, 406854)-Net over F₈ — Digital

Digital (143, 173, 406854)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(173−30, 173, large)-Net in Base 8 — Upper bound on s

There is no (143, 173, large)-net in base 8, because

28 times m-reduction [i] would yield (143, 145, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]