Best Known (54, 54+19, s)-Nets in Base 16
(54, 54+19, 7283)-Net over F16 — Constructive and digital
Digital (54, 73, 7283)-net over F16, using
- 161 times duplication [i] based on digital (53, 72, 7283)-net over F16, using
- net defined by OOA [i] based on linear OOA(1672, 7283, F16, 19, 19) (dual of [(7283, 19), 138305, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1672, 65548, F16, 19) (dual of [65548, 65476, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1672, 65551, F16, 19) (dual of [65551, 65479, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 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(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(1672, 65551, F16, 19) (dual of [65551, 65479, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1672, 65548, F16, 19) (dual of [65548, 65476, 20]-code), using
- net defined by OOA [i] based on linear OOA(1672, 7283, F16, 19, 19) (dual of [(7283, 19), 138305, 20]-NRT-code), using
(54, 54+19, 60200)-Net over F16 — Digital
Digital (54, 73, 60200)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1673, 60200, F16, 19) (dual of [60200, 60127, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1673, 65553, F16, 19) (dual of [65553, 65480, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(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(164, 17, F16, 4) (dual of [17, 13, 5]-code or 17-arc in PG(3,16)), using
- extended Reed–Solomon code RSe(13,16) [i]
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1673, 65553, F16, 19) (dual of [65553, 65480, 20]-code), using
(54, 54+19, large)-Net in Base 16 — Upper bound on s
There is no (54, 73, large)-net in base 16, because
- 17 times m-reduction [i] would yield (54, 56, large)-net in base 16, but