Best Known (68−18, 68, s)-Nets in Base 16
(68−18, 68, 7283)-Net over F16 — Constructive and digital
Digital (50, 68, 7283)-net over F16, using
- net defined by OOA [i] based on linear OOA(1668, 7283, F16, 18, 18) (dual of [(7283, 18), 131026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1668, 65547, F16, 18) (dual of [65547, 65479, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 65551, F16, 18) (dual of [65551, 65483, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(1665, 65536, F16, 18) (dual of [65536, 65471, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1653, 65536, F16, 14) (dual of [65536, 65483, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(163, 15, F16, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,16) or 15-cap in PG(2,16)), using
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- Reed–Solomon code RS(13,16) [i]
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1668, 65551, F16, 18) (dual of [65551, 65483, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(1668, 65547, F16, 18) (dual of [65547, 65479, 19]-code), using
(68−18, 68, 49961)-Net over F16 — Digital
Digital (50, 68, 49961)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1668, 49961, F16, 18) (dual of [49961, 49893, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 65551, F16, 18) (dual of [65551, 65483, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(1665, 65536, F16, 18) (dual of [65536, 65471, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1653, 65536, F16, 14) (dual of [65536, 65483, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(163, 15, F16, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,16) or 15-cap in PG(2,16)), using
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- Reed–Solomon code RS(13,16) [i]
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1668, 65551, F16, 18) (dual of [65551, 65483, 19]-code), using
(68−18, 68, large)-Net in Base 16 — Upper bound on s
There is no (50, 68, large)-net in base 16, because
- 16 times m-reduction [i] would yield (50, 52, large)-net in base 16, but