Best Known (67, 67+24, s)-Nets in Base 16
(67, 67+24, 5462)-Net over F16 — Constructive and digital
Digital (67, 91, 5462)-net over F16, using
- 161 times duplication [i] based on digital (66, 90, 5462)-net over F16, using
- net defined by OOA [i] based on linear OOA(1690, 5462, F16, 24, 24) (dual of [(5462, 24), 130998, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(1690, 65544, F16, 24) (dual of [65544, 65454, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1690, 65545, F16, 24) (dual of [65545, 65455, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- 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(1681, 65536, F16, 22) (dual of [65536, 65455, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(161, 9, F16, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(1690, 65545, F16, 24) (dual of [65545, 65455, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(1690, 65544, F16, 24) (dual of [65544, 65454, 25]-code), using
- net defined by OOA [i] based on linear OOA(1690, 5462, F16, 24, 24) (dual of [(5462, 24), 130998, 25]-NRT-code), using
(67, 67+24, 50887)-Net over F16 — Digital
Digital (67, 91, 50887)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1691, 50887, F16, 24) (dual of [50887, 50796, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1691, 65550, F16, 24) (dual of [65550, 65459, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- 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(1677, 65536, F16, 21) (dual of [65536, 65459, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(1691, 65550, F16, 24) (dual of [65550, 65459, 25]-code), using
(67, 67+24, large)-Net in Base 16 — Upper bound on s
There is no (67, 91, large)-net in base 16, because
- 22 times m-reduction [i] would yield (67, 69, large)-net in base 16, but