MinT - Best Known (55−13, 55, s)-Nets in Base 9

Best Known (55−13, 55, s)-Nets in Base 9

(55−13, 55, 2189)-Net over F₉ — Constructive and digital

Digital (42, 55, 2189)-net over F₉, using

9² times duplication [i] based on digital (40, 53, 2189)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁵³, 2189, F₉, 13, 13) (dual of [(2189, 13), 28404, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(9⁵³, 13135, F₉, 13) (dual of [13135, 13082, 14]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(9⁵², 13134, F₉, 13) (dual of [13134, 13082, 14]-code), using
      - trace code [i] based on linear OA(81²⁶, 6567, F₈₁, 13) (dual of [6567, 6541, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(81²⁵, 6562, F₈₁, 13) (dual of [6562, 6537, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(81²¹, 6562, F₈₁, 11) (dual of [6562, 6541, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(81¹, 5, F₈₁, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(55−13, 55, 15633)-Net over F₉ — Digital

Digital (42, 55, 15633)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−13, 55, large)-Net in Base 9 — Upper bound on s

There is no (42, 55, large)-net in base 9, because

11 times m-reduction [i] would yield (42, 44, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]