MinT - Best Known (46, 67, s)-Nets in Base 27

Best Known (46, 67, s)-Nets in Base 27

(46, 67, 1970)-Net over F₂₇ — Constructive and digital

Digital (46, 67, 1970)-net over F₂₇, using

27² times duplication [i] based on digital (44, 65, 1970)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁶⁵, 1970, F₂₇, 21, 21) (dual of [(1970, 21), 41305, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(27⁶⁵, 19701, F₂₇, 21) (dual of [19701, 19636, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(27⁶⁵, 19702, F₂₇, 21) (dual of [19702, 19637, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
        linear OA(27⁶¹, 19683, F₂₇, 21) (dual of [19683, 19622, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(27⁴⁶, 19683, F₂₇, 16) (dual of [19683, 19637, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(27⁴, 19, F₂₇, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,27)), using
        discarding factors / shortening the dual code based on linear OA(27⁴, 27, F₂₇, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
        Reedâ€“Solomon code RS(23,27) [i]

(46, 67, 19935)-Net over F₂₇ — Digital

Digital (46, 67, 19935)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(46, 67, large)-Net in Base 27 — Upper bound on s

There is no (46, 67, large)-net in base 27, because

19 times m-reduction [i] would yield (46, 48, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]