Best Known (52, 52+27, s)-Nets in Base 32
(52, 52+27, 2520)-Net over F32 — Constructive and digital
Digital (52, 79, 2520)-net over F32, using
- net defined by OOA [i] based on linear OOA(3279, 2520, F32, 27, 27) (dual of [(2520, 27), 67961, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3279, 32761, F32, 27) (dual of [32761, 32682, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3279, 32761, F32, 27) (dual of [32761, 32682, 28]-code), using
(52, 52+27, 16385)-Net over F32 — Digital
Digital (52, 79, 16385)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3279, 16385, F32, 2, 27) (dual of [(16385, 2), 32691, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3279, 32770, F32, 27) (dual of [32770, 32691, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3279, 32771, F32, 27) (dual of [32771, 32692, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3276, 32768, F32, 26) (dual of [32768, 32692, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3279, 32771, F32, 27) (dual of [32771, 32692, 28]-code), using
- OOA 2-folding [i] based on linear OA(3279, 32770, F32, 27) (dual of [32770, 32691, 28]-code), using
(52, 52+27, large)-Net in Base 32 — Upper bound on s
There is no (52, 79, large)-net in base 32, because
- 25 times m-reduction [i] would yield (52, 54, large)-net in base 32, but