Best Known (34, 34+12, s)-Nets in Base 16
(34, 34+12, 21846)-Net over F16 — Constructive and digital
Digital (34, 46, 21846)-net over F16, using
- net defined by OOA [i] based on linear OOA(1646, 21846, F16, 12, 12) (dual of [(21846, 12), 262106, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1646, 131076, F16, 12) (dual of [131076, 131030, 13]-code), using
- trace code [i] based on linear OA(25623, 65538, F256, 12) (dual of [65538, 65515, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(25623, 65538, F256, 12) (dual of [65538, 65515, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1646, 131076, F16, 12) (dual of [131076, 131030, 13]-code), using
(34, 34+12, 79141)-Net over F16 — Digital
Digital (34, 46, 79141)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1646, 79141, F16, 12) (dual of [79141, 79095, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1646, 131072, F16, 12) (dual of [131072, 131026, 13]-code), using
- trace code [i] based on linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- trace code [i] based on linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1646, 131072, F16, 12) (dual of [131072, 131026, 13]-code), using
(34, 34+12, large)-Net in Base 16 — Upper bound on s
There is no (34, 46, large)-net in base 16, because
- 10 times m-reduction [i] would yield (34, 36, large)-net in base 16, but