Best Known (19, 19+7, s)-Nets in Base 16
(19, 19+7, 43691)-Net over F16 — Constructive and digital
Digital (19, 26, 43691)-net over F16, using
- net defined by OOA [i] based on linear OOA(1626, 43691, F16, 7, 7) (dual of [(43691, 7), 305811, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1626, 131074, F16, 7) (dual of [131074, 131048, 8]-code), using
- trace code [i] based on linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- trace code [i] based on linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1626, 131074, F16, 7) (dual of [131074, 131048, 8]-code), using
(19, 19+7, 131076)-Net over F16 — Digital
Digital (19, 26, 131076)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1626, 131076, F16, 7) (dual of [131076, 131050, 8]-code), using
- trace code [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
(19, 19+7, large)-Net in Base 16 — Upper bound on s
There is no (19, 26, large)-net in base 16, because
- 5 times m-reduction [i] would yield (19, 21, large)-net in base 16, but