MinT - Best Known (172−23, 172, s)-Nets in Base 8

Best Known (172−23, 172, s)-Nets in Base 8

(172−23, 172, 762609)-Net over F₈ — Constructive and digital

Digital (149, 172, 762609)-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 (138, 161, 762600)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁶¹, 762600, F₈, 23, 23) (dual of [(762600, 23), 17539639, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(8¹⁶¹, 8388601, F₈, 23) (dual of [8388601, 8388440, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(172−23, 172, large)-Net over F₈ — Digital

Digital (149, 172, large)-net over F₈, using

8⁴ times duplication [i] based on digital (145, 168, large)-net over F₈, using
- t-expansion [i] based on digital (144, 168, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(172−23, 172, large)-Net in Base 8 — Upper bound on s

There is no (149, 172, large)-net in base 8, because

21 times m-reduction [i] would yield (149, 151, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]