MinT - Best Known (34, 49, s)-Nets in Base 27

Best Known (34, 49, s)-Nets in Base 27

(34, 49, 2815)-Net over F₂₇ — Constructive and digital

Digital (34, 49, 2815)-net over F₂₇, using

27¹ times duplication [i] based on digital (33, 48, 2815)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁴⁸, 2815, F₂₇, 15, 15) (dual of [(2815, 15), 42177, 16]-NRT-code), using
  - OOA 7-folding and stacking with additional row [i] based on linear OA(27⁴⁸, 19706, F₂₇, 15) (dual of [19706, 19658, 16]-code), using
    - discarding factors / shortening the dual code based on linear OA(27⁴⁸, 19707, F₂₇, 15) (dual of [19707, 19659, 16]-code), using
      - constructionÂ X applied to C([0,7]) âŠ‚ C([0,4]) [i] based on
        linear OA(27⁴³, 19684, F₂₇, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(27²⁵, 19684, F₂₇, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
        linear OA(27⁵, 23, F₂₇, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,27)), using
        discarding factors / shortening the dual code based on linear OA(27⁵, 27, F₂₇, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
        Reedâ€“Solomon code RS(22,27) [i]

(34, 49, 23789)-Net over F₂₇ — Digital

Digital (34, 49, 23789)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 49, large)-Net in Base 27 — Upper bound on s

There is no (34, 49, large)-net in base 27, because

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