Best Known (27, 27+10, s)-Nets in Base 32
(27, 27+10, 209716)-Net over F32 — Constructive and digital
Digital (27, 37, 209716)-net over F32, using
- net defined by OOA [i] based on linear OOA(3237, 209716, F32, 10, 10) (dual of [(209716, 10), 2097123, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3237, 1048580, F32, 10) (dual of [1048580, 1048543, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3233, 1048576, F32, 9) (dual of [1048576, 1048543, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(3237, 1048580, F32, 10) (dual of [1048580, 1048543, 11]-code), using
(27, 27+10, 720280)-Net over F32 — Digital
Digital (27, 37, 720280)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 720280, F32, 10) (dual of [720280, 720243, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using
(27, 27+10, large)-Net in Base 32 — Upper bound on s
There is no (27, 37, large)-net in base 32, because
- 8 times m-reduction [i] would yield (27, 29, large)-net in base 32, but