Best Known (122−23, 122, s)-Nets in Base 16
(122−23, 122, 95374)-Net over F16 — Constructive and digital
Digital (99, 122, 95374)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- digital (83, 106, 95325)-net over F16, using
- net defined by OOA [i] based on linear OOA(16106, 95325, F16, 23, 23) (dual of [(95325, 23), 2192369, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(16106, 1048576, F16, 23) (dual of [1048576, 1048470, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 11-folding and stacking with additional row [i] based on linear OA(16106, 1048576, F16, 23) (dual of [1048576, 1048470, 24]-code), using
- net defined by OOA [i] based on linear OOA(16106, 95325, F16, 23, 23) (dual of [(95325, 23), 2192369, 24]-NRT-code), using
- digital (5, 16, 49)-net over F16, using
(122−23, 122, 190651)-Net in Base 16 — Constructive
(99, 122, 190651)-net in base 16, using
- 161 times duplication [i] based on (98, 121, 190651)-net in base 16, using
- net defined by OOA [i] based on OOA(16121, 190651, S16, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(16121, 2097162, S16, 23), using
- discarding factors based on OA(16121, 2097163, S16, 23), using
- discarding parts of the base [i] based on linear OA(12869, 2097163, F128, 23) (dual of [2097163, 2097094, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12869, 2097163, F128, 23) (dual of [2097163, 2097094, 24]-code), using
- discarding factors based on OA(16121, 2097163, S16, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(16121, 2097162, S16, 23), using
- net defined by OOA [i] based on OOA(16121, 190651, S16, 23, 23), using
(122−23, 122, 2871812)-Net over F16 — Digital
Digital (99, 122, 2871812)-net over F16, using
(122−23, 122, large)-Net in Base 16 — Upper bound on s
There is no (99, 122, large)-net in base 16, because
- 21 times m-reduction [i] would yield (99, 101, large)-net in base 16, but