MinT - Best Known (98−24, 98, s)-Nets in Base 9

Best Known (98−24, 98, s)-Nets in Base 9

(98−24, 98, 1094)-Net over F₉ — Constructive and digital

Digital (74, 98, 1094)-net over F₉, using

9² times duplication [i] based on digital (72, 96, 1094)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁹⁶, 1094, F₉, 24, 24) (dual of [(1094, 24), 26160, 25]-NRT-code), using
  - OA 12-folding and stacking [i] based on linear OA(9⁹⁶, 13128, F₉, 24) (dual of [13128, 13032, 25]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁹⁶, 13132, F₉, 24) (dual of [13132, 13036, 25]-code), using
      - trace code [i] based on linear OA(81⁴⁸, 6566, F₈₁, 24) (dual of [6566, 6518, 25]-code), using
        constructionÂ X applied to C_e(23) âŠ‚ C_e(21) [i] based on
        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(81⁴³, 6561, F₈₁, 22) (dual of [6561, 6518, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(98−24, 98, 13729)-Net over F₉ — Digital

Digital (74, 98, 13729)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(98−24, 98, large)-Net in Base 9 — Upper bound on s

There is no (74, 98, large)-net in base 9, because

22 times m-reduction [i] would yield (74, 76, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]