MinT - Best Known (109−27, 109, s)-Nets in Base 9

Best Known (109−27, 109, s)-Nets in Base 9

(109−27, 109, 1010)-Net over F₉ — Constructive and digital

Digital (82, 109, 1010)-net over F₉, using

9¹ times duplication [i] based on digital (81, 108, 1010)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹⁰⁸, 1010, F₉, 27, 27) (dual of [(1010, 27), 27162, 28]-NRT-code), using
  - OOA 13-folding and stacking with additional row [i] based on linear OA(9¹⁰⁸, 13131, F₉, 27) (dual of [13131, 13023, 28]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹⁰⁸, 13134, F₉, 27) (dual of [13134, 13026, 28]-code), using
      - trace code [i] based on linear OA(81⁵⁴, 6567, F₈₁, 27) (dual of [6567, 6513, 28]-code), using
        constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
        linear OA(81⁵³, 6562, F₈₁, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(81⁴⁹, 6562, F₈₁, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (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]

(109−27, 109, 13216)-Net over F₉ — Digital

Digital (82, 109, 13216)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−27, 109, large)-Net in Base 9 — Upper bound on s

There is no (82, 109, large)-net in base 9, because

25 times m-reduction [i] would yield (82, 84, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]