Best Known (35−9, 35, s)-Nets in Base 27
(35−9, 35, 132863)-Net over F27 — Constructive and digital
Digital (26, 35, 132863)-net over F27, using
- net defined by OOA [i] based on linear OOA(2735, 132863, F27, 9, 9) (dual of [(132863, 9), 1195732, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2735, 531453, F27, 9) (dual of [531453, 531418, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2735, 531455, F27, 9) (dual of [531455, 531420, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(2733, 531441, F27, 9) (dual of [531441, 531408, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2721, 531441, F27, 6) (dual of [531441, 531420, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(2735, 531455, F27, 9) (dual of [531455, 531420, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2735, 531453, F27, 9) (dual of [531453, 531418, 10]-code), using
(35−9, 35, 531455)-Net over F27 — Digital
Digital (26, 35, 531455)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2735, 531455, F27, 9) (dual of [531455, 531420, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(2733, 531441, F27, 9) (dual of [531441, 531408, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2721, 531441, F27, 6) (dual of [531441, 531420, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
(35−9, 35, large)-Net in Base 27 — Upper bound on s
There is no (26, 35, large)-net in base 27, because
- 7 times m-reduction [i] would yield (26, 28, large)-net in base 27, but