Best Known (81, 81+7, s)-Nets in Base 16
(81, 81+7, large)-Net over F16 — Constructive and digital
Digital (81, 88, large)-net over F16, using
- 3 times m-reduction [i] based on digital (81, 91, large)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 10, 3355440)-net over F16, using
- s-reduction based on digital (7, 10, large)-net over F16, using
- net defined by OOA [i] based on linear OOA(1610, large, F16, 3, 3), using
- appending kth column [i] based on linear OOA(1610, large, F16, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(1610, large, F16, 3) (dual of [large, large−10, 4]-code), using
- appending kth column [i] based on linear OOA(1610, large, F16, 2, 3), using
- net defined by OOA [i] based on linear OOA(1610, large, F16, 3, 3), using
- s-reduction based on digital (7, 10, large)-net over F16, using
- digital (20, 25, 3355440)-net over F16, using
- s-reduction based on digital (20, 25, 4194301)-net over F16, using
- net defined by OOA [i] based on linear OOA(1625, 4194301, F16, 5, 5) (dual of [(4194301, 5), 20971480, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1625, large, F16, 5) (dual of [large, large−25, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(1625, large, F16, 5) (dual of [large, large−25, 6]-code), using
- net defined by OOA [i] based on linear OOA(1625, 4194301, F16, 5, 5) (dual of [(4194301, 5), 20971480, 6]-NRT-code), using
- s-reduction based on digital (20, 25, 4194301)-net over F16, 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 (7, 10, 3355440)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(81, 81+7, large)-Net in Base 16 — Upper bound on s
There is no (81, 88, large)-net in base 16, because
- 5 times m-reduction [i] would yield (81, 83, large)-net in base 16, but