Best Known (89−23, 89, s)-Nets in Base 16
(89−23, 89, 5959)-Net over F16 — Constructive and digital
Digital (66, 89, 5959)-net over F16, using
- 162 times duplication [i] based on digital (64, 87, 5959)-net over F16, using
- net defined by OOA [i] based on linear OOA(1687, 5959, F16, 23, 23) (dual of [(5959, 23), 136970, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(1687, 65550, F16, 23) (dual of [65550, 65463, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- 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(1673, 65536, F16, 20) (dual of [65536, 65463, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(1687, 65550, F16, 23) (dual of [65550, 65463, 24]-code), using
- net defined by OOA [i] based on linear OOA(1687, 5959, F16, 23, 23) (dual of [(5959, 23), 136970, 24]-NRT-code), using
(89−23, 89, 64293)-Net over F16 — Digital
Digital (66, 89, 64293)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1689, 64293, F16, 23) (dual of [64293, 64204, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1689, 65556, F16, 23) (dual of [65556, 65467, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- 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(1669, 65536, F16, 19) (dual of [65536, 65467, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(164, 20, F16, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,16)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(1689, 65556, F16, 23) (dual of [65556, 65467, 24]-code), using
(89−23, 89, large)-Net in Base 16 — Upper bound on s
There is no (66, 89, large)-net in base 16, because
- 21 times m-reduction [i] would yield (66, 68, large)-net in base 16, but