Best Known (64, 64+20, s)-Nets in Base 16
(64, 64+20, 13109)-Net over F16 — Constructive and digital
Digital (64, 84, 13109)-net over F16, using
- net defined by OOA [i] based on linear OOA(1684, 13109, F16, 20, 20) (dual of [(13109, 20), 262096, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1684, 131090, F16, 20) (dual of [131090, 131006, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1684, 131094, F16, 20) (dual of [131094, 131010, 21]-code), using
- trace code [i] based on linear OA(25642, 65547, F256, 20) (dual of [65547, 65505, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(25642, 65547, F256, 20) (dual of [65547, 65505, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1684, 131094, F16, 20) (dual of [131094, 131010, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1684, 131090, F16, 20) (dual of [131090, 131006, 21]-code), using
(64, 64+20, 131094)-Net over F16 — Digital
Digital (64, 84, 131094)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1684, 131094, F16, 20) (dual of [131094, 131010, 21]-code), using
- trace code [i] based on linear OA(25642, 65547, F256, 20) (dual of [65547, 65505, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(25642, 65547, F256, 20) (dual of [65547, 65505, 21]-code), using
(64, 64+20, large)-Net in Base 16 — Upper bound on s
There is no (64, 84, large)-net in base 16, because
- 18 times m-reduction [i] would yield (64, 66, large)-net in base 16, but