MinT - Best Known (106−26, 106, s)-Nets in Base 9

Best Known (106−26, 106, s)-Nets in Base 9

(106−26, 106, 1010)-Net over F₉ — Constructive and digital

Digital (80, 106, 1010)-net over F₉, using

9² times duplication [i] based on digital (78, 104, 1010)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹⁰⁴, 1010, F₉, 26, 26) (dual of [(1010, 26), 26156, 27]-NRT-code), using
  - OA 13-folding and stacking [i] based on linear OA(9¹⁰⁴, 13130, F₉, 26) (dual of [13130, 13026, 27]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹⁰⁴, 13132, F₉, 26) (dual of [13132, 13028, 27]-code), using
      - trace code [i] based on linear OA(81⁵², 6566, F₈₁, 26) (dual of [6566, 6514, 27]-code), using
        constructionÂ X applied to C_e(25) âŠ‚ C_e(23) [i] based on
        linear OA(81⁵¹, 6561, F₈₁, 26) (dual of [6561, 6510, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
        linear OA(81⁴⁷, 6561, F₈₁, 24) (dual of [6561, 6514, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [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]

(106−26, 106, 14155)-Net over F₉ — Digital

Digital (80, 106, 14155)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−26, 106, large)-Net in Base 9 — Upper bound on s

There is no (80, 106, large)-net in base 9, because

24 times m-reduction [i] would yield (80, 82, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]