MinT - Best Known (146−30, 146, s)-Nets in Base 8

Best Known (146−30, 146, s)-Nets in Base 8

(146−30, 146, 2193)-Net over F₈ — Constructive and digital

Digital (116, 146, 2193)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (101, 131, 2184)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³¹, 2184, F₈, 30, 30) (dual of [(2184, 30), 65389, 31]-NRT-code), using
    - OA 15-folding and stacking [i] based on linear OA(8¹³¹, 32760, F₈, 30) (dual of [32760, 32629, 31]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹³¹, 32768, F₈, 30) (dual of [32768, 32637, 31]-code), using
        an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(146−30, 146, 58711)-Net over F₈ — Digital

Digital (116, 146, 58711)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(146−30, 146, large)-Net in Base 8 — Upper bound on s

There is no (116, 146, large)-net in base 8, because

28 times m-reduction [i] would yield (116, 118, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]