Best Known (50, 50+11, s)-Nets in Base 16
(50, 50+11, 1677720)-Net over F16 — Constructive and digital
Digital (50, 61, 1677720)-net over F16, using
- net defined by OOA [i] based on linear OOA(1661, 1677720, F16, 11, 11) (dual of [(1677720, 11), 18454859, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1661, 8388601, F16, 11) (dual of [8388601, 8388540, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(1661, large, F16, 11) (dual of [large, large−61, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(1661, large, F16, 11) (dual of [large, large−61, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1661, 8388601, F16, 11) (dual of [8388601, 8388540, 12]-code), using
(50, 50+11, large)-Net over F16 — Digital
Digital (50, 61, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1661, large, F16, 11) (dual of [large, large−61, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
(50, 50+11, large)-Net in Base 16 — Upper bound on s
There is no (50, 61, large)-net in base 16, because
- 9 times m-reduction [i] would yield (50, 52, large)-net in base 16, but