Best Known (54, 54+28, s)-Nets in Base 32
(54, 54+28, 2340)-Net over F32 — Constructive and digital
Digital (54, 82, 2340)-net over F32, using
- net defined by OOA [i] based on linear OOA(3282, 2340, F32, 28, 28) (dual of [(2340, 28), 65438, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3282, 32760, F32, 28) (dual of [32760, 32678, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 32768, F32, 28) (dual of [32768, 32686, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(3282, 32768, F32, 28) (dual of [32768, 32686, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3282, 32760, F32, 28) (dual of [32760, 32678, 29]-code), using
(54, 54+28, 16623)-Net over F32 — Digital
Digital (54, 82, 16623)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3282, 16623, F32, 28) (dual of [16623, 16541, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 32768, F32, 28) (dual of [32768, 32686, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(3282, 32768, F32, 28) (dual of [32768, 32686, 29]-code), using
(54, 54+28, large)-Net in Base 32 — Upper bound on s
There is no (54, 82, large)-net in base 32, because
- 26 times m-reduction [i] would yield (54, 56, large)-net in base 32, but