Best Known (54, 54+19, s)-Nets in Base 32
(54, 54+19, 116508)-Net over F32 — Constructive and digital
Digital (54, 73, 116508)-net over F32, using
- net defined by OOA [i] based on linear OOA(3273, 116508, F32, 19, 19) (dual of [(116508, 19), 2213579, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3273, 1048573, F32, 19) (dual of [1048573, 1048500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3273, 1048573, F32, 19) (dual of [1048573, 1048500, 20]-code), using
(54, 54+19, 548694)-Net over F32 — Digital
Digital (54, 73, 548694)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3273, 548694, F32, 19) (dual of [548694, 548621, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
(54, 54+19, large)-Net in Base 32 — Upper bound on s
There is no (54, 73, large)-net in base 32, because
- 17 times m-reduction [i] would yield (54, 56, large)-net in base 32, but