Best Known (157−4, 157, s)-Nets in Base 4
(157−4, 157, large)-Net over F4 — Constructive and digital
Digital (153, 157, large)-net over F4, using
- t-expansion [i] based on digital (149, 157, large)-net over F4, using
- 3 times m-reduction [i] based on digital (149, 160, large)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (32, 37, 4194301)-net over F4, using
- net defined by OOA [i] based on linear OOA(437, 4194301, F4, 5, 5) (dual of [(4194301, 5), 20971468, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(437, large, F4, 5) (dual of [large, large−37, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(437, large, F4, 5) (dual of [large, large−37, 6]-code), using
- net defined by OOA [i] based on linear OOA(437, 4194301, F4, 5, 5) (dual of [(4194301, 5), 20971468, 6]-NRT-code), using
- digital (112, 123, 5033160)-net over F4, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- digital (32, 37, 4194301)-net over F4, using
- (u, u+v)-construction [i] based on
- 3 times m-reduction [i] based on digital (149, 160, large)-net over F4, using
(157−4, 157, large)-Net in Base 4 — Upper bound on s
There is no (153, 157, large)-net in base 4, because
- 2 times m-reduction [i] would yield (153, 155, large)-net in base 4, but