MinT - Best Known (59−14, 59, s)-Nets in Base 9

Best Known (59−14, 59, s)-Nets in Base 9

(59−14, 59, 1877)-Net over F₉ — Constructive and digital

Digital (45, 59, 1877)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁵⁹, 1877, F₉, 14, 14) (dual of [(1877, 14), 26219, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(9⁵⁹, 13139, F₉, 14) (dual of [13139, 13080, 15]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(9⁵⁸, 13138, F₉, 14) (dual of [13138, 13080, 15]-code), using
    - trace code [i] based on linear OA(81²⁹, 6569, F₈₁, 14) (dual of [6569, 6540, 15]-code), using
      - constructionÂ X applied to C_e(13) âŠ‚ C_e(10) [i] based on
        linear OA(81²⁷, 6561, F₈₁, 14) (dual of [6561, 6534, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(81²¹, 6561, F₈₁, 11) (dual of [6561, 6540, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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]

(59−14, 59, 15181)-Net over F₉ — Digital

Digital (45, 59, 15181)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−14, 59, large)-Net in Base 9 — Upper bound on s

There is no (45, 59, large)-net in base 9, because

12 times m-reduction [i] would yield (45, 47, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]