Best Known (56−14, 56, s)-Nets in Base 16
(56−14, 56, 18726)-Net over F16 — Constructive and digital
Digital (42, 56, 18726)-net over F16, using
- net defined by OOA [i] based on linear OOA(1656, 18726, F16, 14, 14) (dual of [(18726, 14), 262108, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1656, 131082, F16, 14) (dual of [131082, 131026, 15]-code), using
- trace code [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1656, 131082, F16, 14) (dual of [131082, 131026, 15]-code), using
(56−14, 56, 116448)-Net over F16 — Digital
Digital (42, 56, 116448)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1656, 116448, F16, 14) (dual of [116448, 116392, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1656, 131082, F16, 14) (dual of [131082, 131026, 15]-code), using
- trace code [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1656, 131082, F16, 14) (dual of [131082, 131026, 15]-code), using
(56−14, 56, large)-Net in Base 16 — Upper bound on s
There is no (42, 56, large)-net in base 16, because
- 12 times m-reduction [i] would yield (42, 44, large)-net in base 16, but