Best Known (27−9, 27, s)-Nets in Base 32
(27−9, 27, 8194)-Net over F32 — Constructive and digital
Digital (18, 27, 8194)-net over F32, using
- net defined by OOA [i] based on linear OOA(3227, 8194, F32, 9, 9) (dual of [(8194, 9), 73719, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3227, 32777, F32, 9) (dual of [32777, 32750, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 32779, F32, 9) (dual of [32779, 32752, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(3225, 32768, F32, 9) (dual of [32768, 32743, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3216, 32768, F32, 6) (dual of [32768, 32752, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(322, 11, F32, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 32779, F32, 9) (dual of [32779, 32752, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3227, 32777, F32, 9) (dual of [32777, 32750, 10]-code), using
(27−9, 27, 32779)-Net over F32 — Digital
Digital (18, 27, 32779)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 32779, F32, 9) (dual of [32779, 32752, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(3225, 32768, F32, 9) (dual of [32768, 32743, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3216, 32768, F32, 6) (dual of [32768, 32752, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(322, 11, F32, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
(27−9, 27, large)-Net in Base 32 — Upper bound on s
There is no (18, 27, large)-net in base 32, because
- 7 times m-reduction [i] would yield (18, 20, large)-net in base 32, but