MinT - Best Known (123, 154, s)-Nets in Base 8

Best Known (123, 154, s)-Nets in Base 8

(123, 154, 2208)-Net over F₈ — Constructive and digital

Digital (123, 154, 2208)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 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 (105, 136, 2184)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁶, 2184, F₈, 31, 31) (dual of [(2184, 31), 67568, 32]-NRT-code), using
    - OOA 15-folding and stacking with additional row [i] based on linear OA(8¹³⁶, 32761, F₈, 31) (dual of [32761, 32625, 32]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹³⁶, 32768, F₈, 31) (dual of [32768, 32632, 32]-code), using
        an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(123, 154, 74411)-Net over F₈ — Digital

Digital (123, 154, 74411)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 154, large)-Net in Base 8 — Upper bound on s

There is no (123, 154, large)-net in base 8, because

29 times m-reduction [i] would yield (123, 125, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]