Best Known (86, 86+29, s)-Nets in Base 16
(86, 86+29, 9362)-Net over F16 — Constructive and digital
Digital (86, 115, 9362)-net over F16, using
- 161 times duplication [i] based on digital (85, 114, 9362)-net over F16, using
- net defined by OOA [i] based on linear OOA(16114, 9362, F16, 29, 29) (dual of [(9362, 29), 271384, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16114, 131069, F16, 29) (dual of [131069, 130955, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16114, 131074, F16, 29) (dual of [131074, 130960, 30]-code), using
- trace code [i] based on linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- trace code [i] based on linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16114, 131074, F16, 29) (dual of [131074, 130960, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16114, 131069, F16, 29) (dual of [131069, 130955, 30]-code), using
- net defined by OOA [i] based on linear OOA(16114, 9362, F16, 29, 29) (dual of [(9362, 29), 271384, 30]-NRT-code), using
(86, 86+29, 88371)-Net over F16 — Digital
Digital (86, 115, 88371)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16115, 88371, F16, 29) (dual of [88371, 88256, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16115, 131077, F16, 29) (dual of [131077, 130962, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16114, 131076, F16, 29) (dual of [131076, 130962, 30]-code), using
- trace code [i] based on linear OA(25657, 65538, F256, 29) (dual of [65538, 65481, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- trace code [i] based on linear OA(25657, 65538, F256, 29) (dual of [65538, 65481, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16114, 131076, F16, 29) (dual of [131076, 130962, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16115, 131077, F16, 29) (dual of [131077, 130962, 30]-code), using
(86, 86+29, large)-Net in Base 16 — Upper bound on s
There is no (86, 115, large)-net in base 16, because
- 27 times m-reduction [i] would yield (86, 88, large)-net in base 16, but