Best Known (45−12, 45, s)-Nets in Base 32
(45−12, 45, 174763)-Net over F32 — Constructive and digital
Digital (33, 45, 174763)-net over F32, using
- net defined by OOA [i] based on linear OOA(3245, 174763, F32, 12, 12) (dual of [(174763, 12), 2097111, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3245, 1048578, F32, 12) (dual of [1048578, 1048533, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 1048580, F32, 12) (dual of [1048580, 1048535, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(3241, 1048576, F32, 11) (dual of [1048576, 1048535, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 1048580, F32, 12) (dual of [1048580, 1048535, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3245, 1048578, F32, 12) (dual of [1048578, 1048533, 13]-code), using
(45−12, 45, 612733)-Net over F32 — Digital
Digital (33, 45, 612733)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 612733, F32, 12) (dual of [612733, 612688, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using
(45−12, 45, large)-Net in Base 32 — Upper bound on s
There is no (33, 45, large)-net in base 32, because
- 10 times m-reduction [i] would yield (33, 35, large)-net in base 32, but