Best Known (61, 61+10, s)-Nets in Base 16
(61, 61+10, 3420721)-Net over F16 — Constructive and digital
Digital (61, 71, 3420721)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (10, 15, 65281)-net over F16, using
- net defined by OOA [i] based on linear OOA(1615, 65281, F16, 5, 5) (dual of [(65281, 5), 326390, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1615, 130563, F16, 5) (dual of [130563, 130548, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1615, 130563, F16, 5) (dual of [130563, 130548, 6]-code), using
- net defined by OOA [i] based on linear OOA(1615, 65281, F16, 5, 5) (dual of [(65281, 5), 326390, 6]-NRT-code), using
- digital (46, 56, 3355440)-net over F16, using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- digital (10, 15, 65281)-net over F16, using
(61, 61+10, large)-Net over F16 — Digital
Digital (61, 71, large)-net over F16, using
- t-expansion [i] based on digital (60, 71, large)-net over F16, using
- 2 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
- 2 times m-reduction [i] based on digital (60, 73, large)-net over F16, using
(61, 61+10, large)-Net in Base 16 — Upper bound on s
There is no (61, 71, large)-net in base 16, because
- 8 times m-reduction [i] would yield (61, 63, large)-net in base 16, but