Best Known (32, 32+17, s)-Nets in Base 32
(32, 32+17, 4096)-Net over F32 — Constructive and digital
Digital (32, 49, 4096)-net over F32, using
- net defined by OOA [i] based on linear OOA(3249, 4096, F32, 17, 17) (dual of [(4096, 17), 69583, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using
(32, 32+17, 16385)-Net over F32 — Digital
Digital (32, 49, 16385)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3249, 16385, F32, 2, 17) (dual of [(16385, 2), 32721, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3249, 32770, F32, 17) (dual of [32770, 32721, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 32771, F32, 17) (dual of [32771, 32722, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3249, 32768, F32, 17) (dual of [32768, 32719, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 32771, F32, 17) (dual of [32771, 32722, 18]-code), using
- OOA 2-folding [i] based on linear OA(3249, 32770, F32, 17) (dual of [32770, 32721, 18]-code), using
(32, 32+17, large)-Net in Base 32 — Upper bound on s
There is no (32, 49, large)-net in base 32, because
- 15 times m-reduction [i] would yield (32, 34, large)-net in base 32, but