Best Known (69−9, 69, s)-Nets in Base 16
(69−9, 69, 8388600)-Net over F16 — Constructive and digital
Digital (60, 69, 8388600)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (15, 19, 4194301)-net over F16, using
- net defined by OOA [i] based on linear OOA(1619, 4194301, F16, 4, 4) (dual of [(4194301, 4), 16777185, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1619, 8388602, F16, 4) (dual of [8388602, 8388583, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(1619, large, F16, 4) (dual of [large, large−19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(1619, large, F16, 4) (dual of [large, large−19, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(1619, 8388602, F16, 4) (dual of [8388602, 8388583, 5]-code), using
- net defined by OOA [i] based on linear OOA(1619, 4194301, F16, 4, 4) (dual of [(4194301, 4), 16777185, 5]-NRT-code), using
- digital (41, 50, 4194300)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 4194300, F16, 10, 9) (dual of [(4194300, 10), 41942950, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1650, 8388601, F16, 2, 9) (dual of [(8388601, 2), 16777152, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1650, 8388602, F16, 2, 9) (dual of [(8388602, 2), 16777154, 10]-NRT-code), using
- trace code [i] based on linear OOA(25625, 4194301, F256, 2, 9) (dual of [(4194301, 2), 8388577, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25625, 8388602, F256, 9) (dual of [8388602, 8388577, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- OOA 2-folding [i] based on linear OA(25625, 8388602, F256, 9) (dual of [8388602, 8388577, 10]-code), using
- trace code [i] based on linear OOA(25625, 4194301, F256, 2, 9) (dual of [(4194301, 2), 8388577, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1650, 8388602, F16, 2, 9) (dual of [(8388602, 2), 16777154, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(1650, 8388601, F16, 2, 9) (dual of [(8388601, 2), 16777152, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1650, 4194300, F16, 10, 9) (dual of [(4194300, 10), 41942950, 10]-NRT-code), using
- digital (15, 19, 4194301)-net over F16, using
(69−9, 69, large)-Net over F16 — Digital
Digital (60, 69, large)-net over F16, using
- 4 times m-reduction [i] based on digital (60, 73, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1673, large, F16, 13) (dual of [large, large−73, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1673, large, F16, 13) (dual of [large, large−73, 14]-code), using
(69−9, 69, large)-Net in Base 16 — Upper bound on s
There is no (60, 69, large)-net in base 16, because
- 7 times m-reduction [i] would yield (60, 62, large)-net in base 16, but