Best Known (52, 52+16, s)-Nets in Base 16
(52, 52+16, 16386)-Net over F16 — Constructive and digital
Digital (52, 68, 16386)-net over F16, using
- 162 times duplication [i] based on digital (50, 66, 16386)-net over F16, using
- net defined by OOA [i] based on linear OOA(1666, 16386, F16, 16, 16) (dual of [(16386, 16), 262110, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(1666, 131088, F16, 16) (dual of [131088, 131022, 17]-code), using
- trace code [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- 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(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(1666, 131088, F16, 16) (dual of [131088, 131022, 17]-code), using
- net defined by OOA [i] based on linear OOA(1666, 16386, F16, 16, 16) (dual of [(16386, 16), 262110, 17]-NRT-code), using
(52, 52+16, 131094)-Net over F16 — Digital
Digital (52, 68, 131094)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1668, 131094, F16, 16) (dual of [131094, 131026, 17]-code), using
- trace code [i] based on linear OA(25634, 65547, F256, 16) (dual of [65547, 65513, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(15) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(25634, 65547, F256, 16) (dual of [65547, 65513, 17]-code), using
(52, 52+16, large)-Net in Base 16 — Upper bound on s
There is no (52, 68, large)-net in base 16, because
- 14 times m-reduction [i] would yield (52, 54, large)-net in base 16, but