Best Known (104−27, 104, s)-Nets in Base 16
(104−27, 104, 5042)-Net over F16 — Constructive and digital
Digital (77, 104, 5042)-net over F16, using
- 161 times duplication [i] based on digital (76, 103, 5042)-net over F16, using
- net defined by OOA [i] based on linear OOA(16103, 5042, F16, 27, 27) (dual of [(5042, 27), 136031, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16103, 65547, F16, 27) (dual of [65547, 65444, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16103, 65550, F16, 27) (dual of [65550, 65447, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- 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(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(26) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(16103, 65550, F16, 27) (dual of [65550, 65447, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16103, 65547, F16, 27) (dual of [65547, 65444, 28]-code), using
- net defined by OOA [i] based on linear OOA(16103, 5042, F16, 27, 27) (dual of [(5042, 27), 136031, 28]-NRT-code), using
(104−27, 104, 62004)-Net over F16 — Digital
Digital (77, 104, 62004)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16104, 62004, F16, 27) (dual of [62004, 61900, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16104, 65554, F16, 27) (dual of [65554, 65450, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- 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(1685, 65536, F16, 23) (dual of [65536, 65451, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(16104, 65554, F16, 27) (dual of [65554, 65450, 28]-code), using
(104−27, 104, large)-Net in Base 16 — Upper bound on s
There is no (77, 104, large)-net in base 16, because
- 25 times m-reduction [i] would yield (77, 79, large)-net in base 16, but