Best Known (78, 78+26, s)-Nets in Base 16
(78, 78+26, 10083)-Net over F16 — Constructive and digital
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
(78, 78+26, 96138)-Net over F16 — Digital
Digital (78, 104, 96138)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16104, 96138, F16, 26) (dual of [96138, 96034, 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
(78, 78+26, large)-Net in Base 16 — Upper bound on s
There is no (78, 104, large)-net in base 16, because
- 24 times m-reduction [i] would yield (78, 80, large)-net in base 16, but