Best Known (48, 48+25, s)-Nets in Base 32
(48, 48+25, 2730)-Net over F32 — Constructive and digital
Digital (48, 73, 2730)-net over F32, using
- net defined by OOA [i] based on linear OOA(3273, 2730, F32, 25, 25) (dual of [(2730, 25), 68177, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3273, 32761, F32, 25) (dual of [32761, 32688, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 32768, F32, 25) (dual of [32768, 32695, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(3273, 32768, F32, 25) (dual of [32768, 32695, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3273, 32761, F32, 25) (dual of [32761, 32688, 26]-code), using
(48, 48+25, 16385)-Net over F32 — Digital
Digital (48, 73, 16385)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3273, 16385, F32, 2, 25) (dual of [(16385, 2), 32697, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3273, 32770, F32, 25) (dual of [32770, 32697, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 32771, F32, 25) (dual of [32771, 32698, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(3273, 32768, F32, 25) (dual of [32768, 32695, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3270, 32768, F32, 24) (dual of [32768, 32698, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(3273, 32771, F32, 25) (dual of [32771, 32698, 26]-code), using
- OOA 2-folding [i] based on linear OA(3273, 32770, F32, 25) (dual of [32770, 32697, 26]-code), using
(48, 48+25, large)-Net in Base 32 — Upper bound on s
There is no (48, 73, large)-net in base 32, because
- 23 times m-reduction [i] would yield (48, 50, large)-net in base 32, but