MinT - Best Known (132−23, 132, s)-Nets in Base 8

Best Known (132−23, 132, s)-Nets in Base 8

(132−23, 132, 23840)-Net over F₈ — Constructive and digital

Digital (109, 132, 23840)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 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 (98, 121, 23831)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²¹, 23831, F₈, 23, 23) (dual of [(23831, 23), 547992, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(8¹²¹, 262142, F₈, 23) (dual of [262142, 262021, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²¹, 262144, F₈, 23) (dual of [262144, 262023, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(132−23, 132, 339085)-Net over F₈ — Digital

Digital (109, 132, 339085)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(132−23, 132, large)-Net in Base 8 — Upper bound on s

There is no (109, 132, large)-net in base 8, because

21 times m-reduction [i] would yield (109, 111, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]