Best Known (231, 231+28, s)-Nets in Base 4
(231, 231+28, 599186)-Net over F4 — Constructive and digital
Digital (231, 259, 599186)-net over F4, using
- 42 times duplication [i] based on digital (229, 257, 599186)-net over F4, using
- t-expansion [i] based on digital (228, 257, 599186)-net over F4, using
- net defined by OOA [i] based on linear OOA(4257, 599186, F4, 30, 29) (dual of [(599186, 30), 17975323, 30]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OOA(4257, 4194303, F4, 2, 29) (dual of [(4194303, 2), 8388349, 30]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4253, 4194301, F4, 2, 29) (dual of [(4194301, 2), 8388349, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4253, 8388602, F4, 29) (dual of [8388602, 8388349, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4253, large, F4, 29) (dual of [large, large−253, 30]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(4253, large, F4, 29) (dual of [large, large−253, 30]-code), using
- OOA 2-folding [i] based on linear OA(4253, 8388602, F4, 29) (dual of [8388602, 8388349, 30]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4253, 4194301, F4, 2, 29) (dual of [(4194301, 2), 8388349, 30]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OOA(4257, 4194303, F4, 2, 29) (dual of [(4194303, 2), 8388349, 30]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4257, 599186, F4, 30, 29) (dual of [(599186, 30), 17975323, 30]-NRT-code), using
- t-expansion [i] based on digital (228, 257, 599186)-net over F4, using
(231, 231+28, 4194304)-Net over F4 — Digital
Digital (231, 259, 4194304)-net over F4, using
- 41 times duplication [i] based on digital (230, 258, 4194304)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4258, 4194304, F4, 2, 28) (dual of [(4194304, 2), 8388350, 29]-NRT-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(4252, 4194301, F4, 2, 28) (dual of [(4194301, 2), 8388350, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4252, 8388602, F4, 28) (dual of [8388602, 8388350, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4252, large, F4, 28) (dual of [large, large−252, 29]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4252, large, F4, 28) (dual of [large, large−252, 29]-code), using
- OOA 2-folding [i] based on linear OA(4252, 8388602, F4, 28) (dual of [8388602, 8388350, 29]-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(4252, 4194301, F4, 2, 28) (dual of [(4194301, 2), 8388350, 29]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4258, 4194304, F4, 2, 28) (dual of [(4194304, 2), 8388350, 29]-NRT-code), using
(231, 231+28, large)-Net in Base 4 — Upper bound on s
There is no (231, 259, large)-net in base 4, because
- 26 times m-reduction [i] would yield (231, 233, large)-net in base 4, but