MinT - Best Known (112, 112+17, s)-Nets in Base 9

Best Known (112, 112+17, s)-Nets in Base 9

(112, 112+17, 1048585)-Net over F₉ — Constructive and digital

Digital (112, 129, 1048585)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (104, 121, 1048575)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²¹, 1048575, F₉, 17, 17) (dual of [(1048575, 17), 17825654, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(9¹²¹, 8388601, F₉, 17) (dual of [8388601, 8388480, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹²¹, large, F₉, 17) (dual of [large, large−121, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(112, 112+17, large)-Net over F₉ — Digital

Digital (112, 129, large)-net over F₉, using

t-expansion [i] based on digital (110, 129, large)-net over F₉, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹²⁹, large, F₉, 19) (dual of [large, large−129, 20]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 9¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(112, 112+17, large)-Net in Base 9 — Upper bound on s

There is no (112, 129, large)-net in base 9, because

15 times m-reduction [i] would yield (112, 114, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]