Best Known (31−10, 31, s)-Nets in Base 32
(31−10, 31, 6556)-Net over F32 — Constructive and digital
Digital (21, 31, 6556)-net over F32, using
- net defined by OOA [i] based on linear OOA(3231, 6556, F32, 10, 10) (dual of [(6556, 10), 65529, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3231, 32780, F32, 10) (dual of [32780, 32749, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, 32783, F32, 10) (dual of [32783, 32752, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(3228, 32768, F32, 10) (dual of [32768, 32740, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(3231, 32783, F32, 10) (dual of [32783, 32752, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3231, 32780, F32, 10) (dual of [32780, 32749, 11]-code), using
(31−10, 31, 13107)-Net in Base 32 — Constructive
(21, 31, 13107)-net in base 32, using
- net defined by OOA [i] based on OOA(3231, 13107, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3231, 65535, S32, 10), using
- discarding factors based on OA(3231, 65538, S32, 10), using
- discarding parts of the base [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- discarding factors based on OA(3231, 65538, S32, 10), using
- OA 5-folding and stacking [i] based on OA(3231, 65535, S32, 10), using
(31−10, 31, 32783)-Net over F32 — Digital
Digital (21, 31, 32783)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3231, 32783, F32, 10) (dual of [32783, 32752, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(3228, 32768, F32, 10) (dual of [32768, 32740, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
(31−10, 31, large)-Net in Base 32 — Upper bound on s
There is no (21, 31, large)-net in base 32, because
- 8 times m-reduction [i] would yield (21, 23, large)-net in base 32, but