Best Known (81, 110, s)-Nets in Base 16
(81, 110, 4681)-Net over F16 — Constructive and digital
Digital (81, 110, 4681)-net over F16, using
- 161 times duplication [i] based on digital (80, 109, 4681)-net over F16, using
- net defined by OOA [i] based on linear OOA(16109, 4681, F16, 29, 29) (dual of [(4681, 29), 135640, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16109, 65535, F16, 29) (dual of [65535, 65426, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16109, 65536, F16, 29) (dual of [65536, 65427, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(16109, 65536, F16, 29) (dual of [65536, 65427, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16109, 65535, F16, 29) (dual of [65535, 65426, 30]-code), using
- net defined by OOA [i] based on linear OOA(16109, 4681, F16, 29, 29) (dual of [(4681, 29), 135640, 30]-NRT-code), using
(81, 110, 52878)-Net over F16 — Digital
Digital (81, 110, 52878)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16110, 52878, F16, 29) (dual of [52878, 52768, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16110, 65545, F16, 29) (dual of [65545, 65435, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(16109, 65536, F16, 29) (dual of [65536, 65427, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(16101, 65536, F16, 27) (dual of [65536, 65435, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(16110, 65545, F16, 29) (dual of [65545, 65435, 30]-code), using
(81, 110, large)-Net in Base 16 — Upper bound on s
There is no (81, 110, large)-net in base 16, because
- 27 times m-reduction [i] would yield (81, 83, large)-net in base 16, but