Best Known (21−7, 21, s)-Nets in Base 27
(21−7, 21, 6564)-Net over F27 — Constructive and digital
Digital (14, 21, 6564)-net over F27, using
- net defined by OOA [i] based on linear OOA(2721, 6564, F27, 7, 7) (dual of [(6564, 7), 45927, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2721, 19693, F27, 7) (dual of [19693, 19672, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2721, 19694, F27, 7) (dual of [19694, 19673, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2710, 19683, F27, 4) (dual of [19683, 19673, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-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(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2721, 19694, F27, 7) (dual of [19694, 19673, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2721, 19693, F27, 7) (dual of [19693, 19672, 8]-code), using
(21−7, 21, 19694)-Net over F27 — Digital
Digital (14, 21, 19694)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2721, 19694, F27, 7) (dual of [19694, 19673, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2710, 19683, F27, 4) (dual of [19683, 19673, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-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(6) ⊂ Ce(3) [i] based on
(21−7, 21, large)-Net in Base 27 — Upper bound on s
There is no (14, 21, large)-net in base 27, because
- 5 times m-reduction [i] would yield (14, 16, large)-net in base 27, but