Best Known (82, 82+19, s)-Nets in Base 16
(82, 82+19, 116573)-Net over F16 — Constructive and digital
Digital (82, 101, 116573)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (67, 86, 116508)-net over F16, using
- net defined by OOA [i] based on linear OOA(1686, 116508, F16, 19, 19) (dual of [(116508, 19), 2213566, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1686, 1048573, F16, 19) (dual of [1048573, 1048487, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1686, 1048573, F16, 19) (dual of [1048573, 1048487, 20]-code), using
- net defined by OOA [i] based on linear OOA(1686, 116508, F16, 19, 19) (dual of [(116508, 19), 2213566, 20]-NRT-code), using
- digital (6, 15, 65)-net over F16, using
(82, 82+19, 233018)-Net in Base 16 — Constructive
(82, 101, 233018)-net in base 16, using
- 161 times duplication [i] based on (81, 100, 233018)-net in base 16, using
- net defined by OOA [i] based on OOA(16100, 233018, S16, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(16100, 2097163, S16, 19), using
- discarding parts of the base [i] based on linear OA(12857, 2097163, F128, 19) (dual of [2097163, 2097106, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12857, 2097163, F128, 19) (dual of [2097163, 2097106, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on OA(16100, 2097163, S16, 19), using
- net defined by OOA [i] based on OOA(16100, 233018, S16, 19, 19), using
(82, 82+19, 2874029)-Net over F16 — Digital
Digital (82, 101, 2874029)-net over F16, using
(82, 82+19, large)-Net in Base 16 — Upper bound on s
There is no (82, 101, large)-net in base 16, because
- 17 times m-reduction [i] would yield (82, 84, large)-net in base 16, but