MinT - Best Known (82, 82+26, s)-Nets in Base 9

Best Known (82, 82+26, s)-Nets in Base 9

(82, 82+26, 1011)-Net over F₉ — Constructive and digital

Digital (82, 108, 1011)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹⁰⁸, 1011, F₉, 26, 26) (dual of [(1011, 26), 26178, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(9¹⁰⁸, 13143, F₉, 26) (dual of [13143, 13035, 27]-code), using
  - discarding factors / shortening the dual code based on linear OA(9¹⁰⁸, 13144, F₉, 26) (dual of [13144, 13036, 27]-code), using
    - trace code [i] based on linear OA(81⁵⁴, 6572, F₈₁, 26) (dual of [6572, 6518, 27]-code), using
      - constructionÂ X applied to C_e(25) âŠ‚ C_e(21) [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(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(81³, 11, F₈₁, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
        discarding factors / shortening the dual code based on linear OA(81³, 81, F₈₁, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
        Reedâ€“Solomon code RS(78,81) [i]

(82, 82+26, 16873)-Net over F₉ — Digital

Digital (82, 108, 16873)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+26, large)-Net in Base 9 — Upper bound on s

There is no (82, 108, large)-net in base 9, because

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