Best Known (68−17, 68, s)-Nets in Base 16
(68−17, 68, 16385)-Net over F16 — Constructive and digital
Digital (51, 68, 16385)-net over F16, using
- net defined by OOA [i] based on linear OOA(1668, 16385, F16, 17, 17) (dual of [(16385, 17), 278477, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1668, 131081, F16, 17) (dual of [131081, 131013, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 131084, F16, 17) (dual of [131084, 131016, 18]-code), using
- trace code [i] based on linear OA(25634, 65542, F256, 17) (dual of [65542, 65508, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(25634, 65542, F256, 17) (dual of [65542, 65508, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 131084, F16, 17) (dual of [131084, 131016, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1668, 131081, F16, 17) (dual of [131081, 131013, 18]-code), using
(68−17, 68, 102340)-Net over F16 — Digital
Digital (51, 68, 102340)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1668, 102340, F16, 17) (dual of [102340, 102272, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 131084, F16, 17) (dual of [131084, 131016, 18]-code), using
- trace code [i] based on linear OA(25634, 65542, F256, 17) (dual of [65542, 65508, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(25634, 65542, F256, 17) (dual of [65542, 65508, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 131084, F16, 17) (dual of [131084, 131016, 18]-code), using
(68−17, 68, large)-Net in Base 16 — Upper bound on s
There is no (51, 68, large)-net in base 16, because
- 15 times m-reduction [i] would yield (51, 53, large)-net in base 16, but