Best Known (25−9, 25, s)-Nets in Base 32
(25−9, 25, 8192)-Net over F32 — Constructive and digital
Digital (16, 25, 8192)-net over F32, using
- net defined by OOA [i] based on linear OOA(3225, 8192, F32, 9, 9) (dual of [(8192, 9), 73703, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using
(25−9, 25, 16385)-Net over F32 — Digital
Digital (16, 25, 16385)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 16385, F32, 2, 9) (dual of [(16385, 2), 32745, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3225, 32770, F32, 9) (dual of [32770, 32745, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 32771, F32, 9) (dual of [32771, 32746, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [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(3222, 32768, F32, 8) (dual of [32768, 32746, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(3225, 32771, F32, 9) (dual of [32771, 32746, 10]-code), using
- OOA 2-folding [i] based on linear OA(3225, 32770, F32, 9) (dual of [32770, 32745, 10]-code), using
(25−9, 25, large)-Net in Base 32 — Upper bound on s
There is no (16, 25, large)-net in base 32, because
- 7 times m-reduction [i] would yield (16, 18, large)-net in base 32, but