MinT - Best Known (142−19, 142, s)-Nets in Base 9

Best Known (142−19, 142, s)-Nets in Base 9

(142−19, 142, 932096)-Net over F₉ — Constructive and digital

Digital (123, 142, 932096)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (110, 129, 932066)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²⁹, 932066, F₉, 19, 19) (dual of [(932066, 19), 17709125, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(9¹²⁹, 8388595, F₉, 19) (dual of [8388595, 8388466, 20]-code), using
      - discarding factors / shortening the dual code 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]

(142−19, 142, large)-Net over F₉ — Digital

Digital (123, 142, large)-net over F₉, using

9⁵ times duplication [i] based on digital (118, 137, large)-net over F₉, using
- t-expansion [i] based on digital (117, 137, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³⁷, large, F₉, 20) (dual of [large, large−137, 21]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(142−19, 142, large)-Net in Base 9 — Upper bound on s

There is no (123, 142, large)-net in base 9, because

17 times m-reduction [i] would yield (123, 125, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]