MinT - Best Known (78−24, 78, s)-Nets in Base 27

Best Known (78−24, 78, s)-Nets in Base 27

(78−24, 78, 1642)-Net over F₂₇ — Constructive and digital

Digital (54, 78, 1642)-net over F₂₇, using

t-expansion [i] based on digital (53, 78, 1642)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁷⁸, 1642, F₂₇, 25, 25) (dual of [(1642, 25), 40972, 26]-NRT-code), using
  - OOA 12-folding and stacking with additional row [i] based on linear OA(27⁷⁸, 19705, F₂₇, 25) (dual of [19705, 19627, 26]-code), using
    - discarding factors / shortening the dual code based on linear OA(27⁷⁸, 19707, F₂₇, 25) (dual of [19707, 19629, 26]-code), using
      - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
        linear OA(27⁷³, 19684, F₂₇, 25) (dual of [19684, 19611, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
        linear OA(27⁵⁵, 19684, F₂₇, 19) (dual of [19684, 19629, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (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]

(78−24, 78, 25934)-Net over F₂₇ — Digital

Digital (54, 78, 25934)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−24, 78, large)-Net in Base 27 — Upper bound on s

There is no (54, 78, large)-net in base 27, because

22 times m-reduction [i] would yield (54, 56, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]