MinT - Best Known (78, 102, s)-Nets in Base 27

Best Known (78, 102, s)-Nets in Base 27

(78, 102, 44290)-Net over F₂₇ — Constructive and digital

Digital (78, 102, 44290)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27¹⁰², 44290, F₂₇, 24, 24) (dual of [(44290, 24), 1062858, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(27¹⁰², 531480, F₂₇, 24) (dual of [531480, 531378, 25]-code), using
  - discarding factors / shortening the dual code based on linear OA(27¹⁰², 531482, F₂₇, 24) (dual of [531482, 531380, 25]-code), using
    - constructionÂ X applied to C_e(23) âŠ‚ C_e(15) [i] based on
      1. linear OA(27⁹³, 531441, F₂₇, 24) (dual of [531441, 531348, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
      2. linear OA(27⁶¹, 531441, F₂₇, 16) (dual of [531441, 531380, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(27⁹, 41, F₂₇, 7) (dual of [41, 32, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁹, 48, F₂₇, 7) (dual of [48, 39, 8]-code), using
        extended algebraic-geometric code AG_e(F,40P) [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]

(78, 102, 807749)-Net over F₂₇ — Digital

Digital (78, 102, 807749)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 102, large)-Net in Base 27 — Upper bound on s

There is no (78, 102, large)-net in base 27, because

22 times m-reduction [i] would yield (78, 80, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]