Best Known (82, 82+28, s)-Nets in Base 32
(82, 82+28, 74898)-Net over F32 — Constructive and digital
Digital (82, 110, 74898)-net over F32, using
- 321 times duplication [i] based on digital (81, 109, 74898)-net over F32, using
- net defined by OOA [i] based on linear OOA(32109, 74898, F32, 28, 28) (dual of [(74898, 28), 2097035, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(32109, 1048572, F32, 28) (dual of [1048572, 1048463, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(32109, 1048576, F32, 28) (dual of [1048576, 1048467, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(32109, 1048576, F32, 28) (dual of [1048576, 1048467, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(32109, 1048572, F32, 28) (dual of [1048572, 1048463, 29]-code), using
- net defined by OOA [i] based on linear OOA(32109, 74898, F32, 28, 28) (dual of [(74898, 28), 2097035, 29]-NRT-code), using
(82, 82+28, 694985)-Net over F32 — Digital
Digital (82, 110, 694985)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32110, 694985, F32, 28) (dual of [694985, 694875, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(32110, 1048585, F32, 28) (dual of [1048585, 1048475, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(32109, 1048576, F32, 28) (dual of [1048576, 1048467, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(32101, 1048576, F32, 26) (dual of [1048576, 1048475, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(32110, 1048585, F32, 28) (dual of [1048585, 1048475, 29]-code), using
(82, 82+28, large)-Net in Base 32 — Upper bound on s
There is no (82, 110, large)-net in base 32, because
- 26 times m-reduction [i] would yield (82, 84, large)-net in base 32, but