MinT - Best Known (52−18, 52, s)-Nets in Base 27

Best Known (52−18, 52, s)-Nets in Base 27

(52−18, 52, 2187)-Net over F₂₇ — Constructive and digital

Digital (34, 52, 2187)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27⁵², 2187, F₂₇, 18, 18) (dual of [(2187, 18), 39314, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(27⁵², 19683, F₂₇, 18) (dual of [19683, 19631, 19]-code), using
  - an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(52−18, 52, 9843)-Net over F₂₇ — Digital

Digital (34, 52, 9843)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(27⁵², 9843, F₂₇, 2, 18) (dual of [(9843, 2), 19634, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(27⁵², 19686, F₂₇, 18) (dual of [19686, 19634, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(27⁵², 19683, F₂₇, 18) (dual of [19683, 19631, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(27⁴⁹, 19683, F₂₇, 17) (dual of [19683, 19634, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(27⁰, 3, F₂₇, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(52−18, 52, large)-Net in Base 27 — Upper bound on s

There is no (34, 52, large)-net in base 27, because

16 times m-reduction [i] would yield (34, 36, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]