Best Known (78, 78+28, s)-Nets in Base 16
(78, 78+28, 4681)-Net over F16 — Constructive and digital
Digital (78, 106, 4681)-net over F16, using
- 161 times duplication [i] based on digital (77, 105, 4681)-net over F16, using
- net defined by OOA [i] based on linear OOA(16105, 4681, F16, 28, 28) (dual of [(4681, 28), 130963, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(16105, 65534, F16, 28) (dual of [65534, 65429, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16105, 65536, F16, 28) (dual of [65536, 65431, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(16105, 65536, F16, 28) (dual of [65536, 65431, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(16105, 65534, F16, 28) (dual of [65534, 65429, 29]-code), using
- net defined by OOA [i] based on linear OOA(16105, 4681, F16, 28, 28) (dual of [(4681, 28), 130963, 29]-NRT-code), using
(78, 78+28, 51272)-Net over F16 — Digital
Digital (78, 106, 51272)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16106, 51272, F16, 28) (dual of [51272, 51166, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16106, 65545, F16, 28) (dual of [65545, 65439, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(16105, 65536, F16, 28) (dual of [65536, 65431, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(1697, 65536, F16, 26) (dual of [65536, 65439, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(161, 9, F16, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 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(16106, 65545, F16, 28) (dual of [65545, 65439, 29]-code), using
(78, 78+28, large)-Net in Base 16 — Upper bound on s
There is no (78, 106, large)-net in base 16, because
- 26 times m-reduction [i] would yield (78, 80, large)-net in base 16, but