Best Known (83, 83+29, s)-Nets in Base 16
(83, 83+29, 4682)-Net over F16 — Constructive and digital
Digital (83, 112, 4682)-net over F16, using
- 161 times duplication [i] based on digital (82, 111, 4682)-net over F16, using
- net defined by OOA [i] based on linear OOA(16111, 4682, F16, 29, 29) (dual of [(4682, 29), 135667, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16111, 65549, F16, 29) (dual of [65549, 65438, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16111, 65550, F16, 29) (dual of [65550, 65439, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [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(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(162, 14, F16, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(16111, 65550, F16, 29) (dual of [65550, 65439, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16111, 65549, F16, 29) (dual of [65549, 65438, 30]-code), using
- net defined by OOA [i] based on linear OOA(16111, 4682, F16, 29, 29) (dual of [(4682, 29), 135667, 30]-NRT-code), using
(83, 83+29, 64937)-Net over F16 — Digital
Digital (83, 112, 64937)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16112, 64937, F16, 29) (dual of [64937, 64825, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16112, 65554, F16, 29) (dual of [65554, 65442, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [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(1693, 65536, F16, 25) (dual of [65536, 65443, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(16112, 65554, F16, 29) (dual of [65554, 65442, 30]-code), using
(83, 83+29, large)-Net in Base 16 — Upper bound on s
There is no (83, 112, large)-net in base 16, because
- 27 times m-reduction [i] would yield (83, 85, large)-net in base 16, but