MinT - Best Known (72, 72+32, s)-Nets in Base 27

Best Known (72, 72+32, s)-Nets in Base 27

(72, 72+32, 1233)-Net over F₂₇ — Constructive and digital

Digital (72, 104, 1233)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27¹⁰⁴, 1233, F₂₇, 32, 32) (dual of [(1233, 32), 39352, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(27¹⁰⁴, 19728, F₂₇, 32) (dual of [19728, 19624, 33]-code), using
  - discarding factors / shortening the dual code based on linear OA(27¹⁰⁴, 19729, F₂₇, 32) (dual of [19729, 19625, 33]-code), using
    - constructionÂ X applied to C_e(31) âŠ‚ C_e(19) [i] based on
      1. linear OA(27⁹¹, 19683, F₂₇, 32) (dual of [19683, 19592, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
      2. linear OA(27⁵⁸, 19683, F₂₇, 20) (dual of [19683, 19625, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
      3. linear OA(27¹³, 46, F₂₇, 11) (dual of [46, 33, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹³, 48, F₂₇, 11) (dual of [48, 35, 12]-code), using
        extended algebraic-geometric code AG_e(F,36P) [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]

(72, 72+32, 30288)-Net over F₂₇ — Digital

Digital (72, 104, 30288)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+32, large)-Net in Base 27 — Upper bound on s

There is no (72, 104, large)-net in base 27, because

30 times m-reduction [i] would yield (72, 74, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]