Best Known (80, 80+26, s)-Nets in Base 16
(80, 80+26, 10083)-Net over F16 — Constructive and digital
Digital (80, 106, 10083)-net over F16, using
- 162 times duplication [i] based on digital (78, 104, 10083)-net over F16, using
- net defined by OOA [i] based on linear OOA(16104, 10083, F16, 26, 26) (dual of [(10083, 26), 262054, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(16104, 131079, F16, 26) (dual of [131079, 130975, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16104, 131082, F16, 26) (dual of [131082, 130978, 27]-code), using
- trace code [i] based on linear OA(25652, 65541, F256, 26) (dual of [65541, 65489, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(25652, 65541, F256, 26) (dual of [65541, 65489, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16104, 131082, F16, 26) (dual of [131082, 130978, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(16104, 131079, F16, 26) (dual of [131079, 130975, 27]-code), using
- net defined by OOA [i] based on linear OOA(16104, 10083, F16, 26, 26) (dual of [(10083, 26), 262054, 27]-NRT-code), using
(80, 80+26, 121130)-Net over F16 — Digital
Digital (80, 106, 121130)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16106, 121130, F16, 26) (dual of [121130, 121024, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16106, 131088, F16, 26) (dual of [131088, 130982, 27]-code), using
- trace code [i] based on linear OA(25653, 65544, F256, 26) (dual of [65544, 65491, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- trace code [i] based on linear OA(25653, 65544, F256, 26) (dual of [65544, 65491, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16106, 131088, F16, 26) (dual of [131088, 130982, 27]-code), using
(80, 80+26, large)-Net in Base 16 — Upper bound on s
There is no (80, 106, large)-net in base 16, because
- 24 times m-reduction [i] would yield (80, 82, large)-net in base 16, but