Best Known (21, 21+7, s)-Nets in Base 16
(21, 21+7, 43694)-Net over F16 — Constructive and digital
Digital (21, 28, 43694)-net over F16, using
- net defined by OOA [i] based on linear OOA(1628, 43694, F16, 7, 7) (dual of [(43694, 7), 305830, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1628, 131083, F16, 7) (dual of [131083, 131055, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1628, 131084, F16, 7) (dual of [131084, 131056, 8]-code), using
- trace code [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [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]
- linear OA(2569, 65537, F256, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [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 C([0,3]) ⊂ C([0,2]) [i] based on
- trace code [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1628, 131084, F16, 7) (dual of [131084, 131056, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1628, 131083, F16, 7) (dual of [131083, 131055, 8]-code), using
(21, 21+7, 131084)-Net over F16 — Digital
Digital (21, 28, 131084)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1628, 131084, F16, 7) (dual of [131084, 131056, 8]-code), using
- trace code [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [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]
- linear OA(2569, 65537, F256, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [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 C([0,3]) ⊂ C([0,2]) [i] based on
- trace code [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
(21, 21+7, large)-Net in Base 16 — Upper bound on s
There is no (21, 28, large)-net in base 16, because
- 5 times m-reduction [i] would yield (21, 23, large)-net in base 16, but