MinT - Best Known (70, 101, s)-Nets in Base 27

Best Known (70, 101, s)-Nets in Base 27

(70, 101, 1315)-Net over F₂₇ — Constructive and digital

Digital (70, 101, 1315)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27¹⁰¹, 1315, F₂₇, 31, 31) (dual of [(1315, 31), 40664, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(27¹⁰¹, 19726, F₂₇, 31) (dual of [19726, 19625, 32]-code), using
  - discarding factors / shortening the dual code based on linear OA(27¹⁰¹, 19729, F₂₇, 31) (dual of [19729, 19628, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(18) [i] based on
      1. linear OA(27⁸⁸, 19683, F₂₇, 31) (dual of [19683, 19595, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(27⁵⁵, 19683, F₂₇, 19) (dual of [19683, 19628, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(70, 101, 30546)-Net over F₂₇ — Digital

Digital (70, 101, 30546)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 101, large)-Net in Base 27 — Upper bound on s

There is no (70, 101, large)-net in base 27, because

29 times m-reduction [i] would yield (70, 72, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]