Best Known (18−6, 18, s)-Nets in Base 32
(18−6, 18, 10926)-Net over F32 — Constructive and digital
Digital (12, 18, 10926)-net over F32, using
- net defined by OOA [i] based on linear OOA(3218, 10926, F32, 6, 6) (dual of [(10926, 6), 65538, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3218, 32778, F32, 6) (dual of [32778, 32760, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 32779, F32, 6) (dual of [32779, 32761, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(327, 32768, F32, 3) (dual of [32768, 32761, 4]-code or 32768-cap in PG(6,32)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(3218, 32779, F32, 6) (dual of [32779, 32761, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(3218, 32778, F32, 6) (dual of [32778, 32760, 7]-code), using
(18−6, 18, 21846)-Net in Base 32 — Constructive
(12, 18, 21846)-net in base 32, using
- net defined by OOA [i] based on OOA(3218, 21846, S32, 6, 6), using
- OA 3-folding and stacking [i] based on OA(3218, 65538, S32, 6), using
- discarding parts of the base [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding parts of the base [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- OA 3-folding and stacking [i] based on OA(3218, 65538, S32, 6), using
(18−6, 18, 32779)-Net over F32 — Digital
Digital (12, 18, 32779)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3218, 32779, F32, 6) (dual of [32779, 32761, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(327, 32768, F32, 3) (dual of [32768, 32761, 4]-code or 32768-cap in PG(6,32)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(5) ⊂ Ce(2) [i] based on
(18−6, 18, large)-Net in Base 32 — Upper bound on s
There is no (12, 18, large)-net in base 32, because
- 4 times m-reduction [i] would yield (12, 14, large)-net in base 32, but