Best Known (77, 77+25, s)-Nets in Base 16
(77, 77+25, 10923)-Net over F16 — Constructive and digital
Digital (77, 102, 10923)-net over F16, using
- 163 times duplication [i] based on digital (74, 99, 10923)-net over F16, using
- net defined by OOA [i] based on linear OOA(1699, 10923, F16, 25, 25) (dual of [(10923, 25), 272976, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1699, 131077, F16, 25) (dual of [131077, 130978, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1698, 131076, F16, 25) (dual of [131076, 130978, 26]-code), using
- trace code [i] based on linear OA(25649, 65538, F256, 25) (dual of [65538, 65489, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(25649, 65538, F256, 25) (dual of [65538, 65489, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1698, 131076, F16, 25) (dual of [131076, 130978, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1699, 131077, F16, 25) (dual of [131077, 130978, 26]-code), using
- net defined by OOA [i] based on linear OOA(1699, 10923, F16, 25, 25) (dual of [(10923, 25), 272976, 26]-NRT-code), using
(77, 77+25, 121893)-Net over F16 — Digital
Digital (77, 102, 121893)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16102, 121893, F16, 25) (dual of [121893, 121791, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(16102, 131088, F16, 25) (dual of [131088, 130986, 26]-code), using
- trace code [i] based on linear OA(25651, 65544, F256, 25) (dual of [65544, 65493, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(25651, 65544, F256, 25) (dual of [65544, 65493, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(16102, 131088, F16, 25) (dual of [131088, 130986, 26]-code), using
(77, 77+25, large)-Net in Base 16 — Upper bound on s
There is no (77, 102, large)-net in base 16, because
- 23 times m-reduction [i] would yield (77, 79, large)-net in base 16, but