MinT - Best Known (90−22, 90, s)-Nets in Base 9

Best Known (90−22, 90, s)-Nets in Base 9

(90−22, 90, 1194)-Net over F₉ — Constructive and digital

Digital (68, 90, 1194)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁹⁰, 1194, F₉, 22, 22) (dual of [(1194, 22), 26178, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(9⁹⁰, 13134, F₉, 22) (dual of [13134, 13044, 23]-code), using
  - discarding factors / shortening the dual code based on linear OA(9⁹⁰, 13138, F₉, 22) (dual of [13138, 13048, 23]-code), using
    - trace code [i] based on linear OA(81⁴⁵, 6569, F₈₁, 22) (dual of [6569, 6524, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(18) [i] based on
        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(81³⁷, 6561, F₈₁, 19) (dual of [6561, 6524, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(81², 8, F₈₁, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
        discarding factors / shortening the dual code based on linear OA(81², 81, F₈₁, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
        Reedâ€“Solomon code RS(79,81) [i]

(90−22, 90, 13346)-Net over F₉ — Digital

Digital (68, 90, 13346)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−22, 90, large)-Net in Base 9 — Upper bound on s

There is no (68, 90, large)-net in base 9, because

20 times m-reduction [i] would yield (68, 70, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]