MinT - Best Known (142−18, 142, s)-Nets in Base 9

Best Known (142−18, 142, s)-Nets in Base 9

(142−18, 142, 1864134)-Net over F₉ — Constructive and digital

Digital (124, 142, 1864134)-net over F₉, using

9¹ times duplication [i] based on digital (123, 141, 1864134)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹⁴¹, 1864134, F₉, 21, 18) (dual of [(1864134, 21), 39146673, 19]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OOA(9¹⁴¹, 5592403, F₉, 3, 18) (dual of [(5592403, 3), 16777068, 19]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(9¹³⁸, 5592402, F₉, 3, 18) (dual of [(5592402, 3), 16777068, 19]-NRT-code), using
      - trace code [i] based on linear OOA(81⁶⁹, 2796201, F₈₁, 3, 18) (dual of [(2796201, 3), 8388534, 19]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81⁶⁹, large, F₈₁, 18) (dual of [large, large−69, 19]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(142−18, 142, large)-Net over F₉ — Digital

Digital (124, 142, large)-net over F₉, using

3 times m-reduction [i] based on digital (124, 145, large)-net over F₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(142−18, 142, large)-Net in Base 9 — Upper bound on s

There is no (124, 142, large)-net in base 9, because

16 times m-reduction [i] would yield (124, 126, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]