MinT - Best Known (60, 60+23, s)-Nets in Base 9

Best Known (60, 60+23, s)-Nets in Base 9

(60, 60+23, 740)-Net over F₉ — Constructive and digital

Digital (60, 83, 740)-net over F₉, using

5 times m-reduction [i] based on digital (60, 88, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 44, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(60, 60+23, 5762)-Net over F₉ — Digital

Digital (60, 83, 5762)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸³, 5762, F₉, 23) (dual of [5762, 5679, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁸³, 6572, F₉, 23) (dual of [6572, 6489, 24]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(9⁸², 6571, F₉, 23) (dual of [6571, 6489, 24]-code), using
    - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
      1. linear OA(9⁸¹, 6562, F₉, 23) (dual of [6562, 6481, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
      2. linear OA(9⁷³, 6562, F₉, 21) (dual of [6562, 6489, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      3. linear OA(9¹, 9, F₉, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(60, 60+23, 7968342)-Net in Base 9 — Upper bound on s

There is no (60, 83, 7968343)-net in base 9, because

1 times m-reduction [i] would yield (60, 82, 7968343)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 769645 119648 187084 355013 806940 527349 175705 735418 519912 068402 800087 007749 660905 > 9⁸² [i]