Best Known (50, 67, s)-Nets in Base 27
(50, 67, 66431)-Net over F27 — Constructive and digital
Digital (50, 67, 66431)-net over F27, using
- 271 times duplication [i] based on digital (49, 66, 66431)-net over F27, using
- net defined by OOA [i] based on linear OOA(2766, 66431, F27, 17, 17) (dual of [(66431, 17), 1129261, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2766, 531449, F27, 17) (dual of [531449, 531383, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2766, 531451, F27, 17) (dual of [531451, 531385, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(2765, 531442, F27, 17) (dual of [531442, 531377, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2757, 531442, F27, 15) (dual of [531442, 531385, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2766, 531451, F27, 17) (dual of [531451, 531385, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2766, 531449, F27, 17) (dual of [531449, 531383, 18]-code), using
- net defined by OOA [i] based on linear OOA(2766, 66431, F27, 17, 17) (dual of [(66431, 17), 1129261, 18]-NRT-code), using
(50, 67, 490668)-Net over F27 — Digital
Digital (50, 67, 490668)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2767, 490668, F27, 17) (dual of [490668, 490601, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2767, 531455, F27, 17) (dual of [531455, 531388, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(2765, 531441, F27, 17) (dual of [531441, 531376, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2753, 531441, F27, 14) (dual of [531441, 531388, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2767, 531455, F27, 17) (dual of [531455, 531388, 18]-code), using
(50, 67, large)-Net in Base 27 — Upper bound on s
There is no (50, 67, large)-net in base 27, because
- 15 times m-reduction [i] would yield (50, 52, large)-net in base 27, but