Best Known (25, 25+9, s)-Nets in Base 16
(25, 25+9, 32768)-Net over F16 — Constructive and digital
Digital (25, 34, 32768)-net over F16, using
- net defined by OOA [i] based on linear OOA(1634, 32768, F16, 9, 9) (dual of [(32768, 9), 294878, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1634, 131073, F16, 9) (dual of [131073, 131039, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 131074, F16, 9) (dual of [131074, 131040, 10]-code), using
- trace code [i] based on linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- trace code [i] based on linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 131074, F16, 9) (dual of [131074, 131040, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1634, 131073, F16, 9) (dual of [131073, 131039, 10]-code), using
(25, 25+9, 106999)-Net over F16 — Digital
Digital (25, 34, 106999)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1634, 106999, F16, 9) (dual of [106999, 106965, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 131074, F16, 9) (dual of [131074, 131040, 10]-code), using
- trace code [i] based on linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- trace code [i] based on linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 131074, F16, 9) (dual of [131074, 131040, 10]-code), using
(25, 25+9, large)-Net in Base 16 — Upper bound on s
There is no (25, 34, large)-net in base 16, because
- 7 times m-reduction [i] would yield (25, 27, large)-net in base 16, but