Best Known (253−28, 253, s)-Nets in Base 4
(253−28, 253, 599185)-Net over F4 — Constructive and digital
Digital (225, 253, 599185)-net over F4, using
- t-expansion [i] based on digital (224, 253, 599185)-net over F4, using
- net defined by OOA [i] based on linear OOA(4253, 599185, F4, 29, 29) (dual of [(599185, 29), 17376112, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4253, 8388591, F4, 29) (dual of [8388591, 8388338, 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 14-folding and stacking with additional row [i] based on linear OA(4253, 8388591, F4, 29) (dual of [8388591, 8388338, 30]-code), using
- net defined by OOA [i] based on linear OOA(4253, 599185, F4, 29, 29) (dual of [(599185, 29), 17376112, 30]-NRT-code), using
(253−28, 253, 3759972)-Net over F4 — Digital
Digital (225, 253, 3759972)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4253, 3759972, F4, 2, 28) (dual of [(3759972, 2), 7519691, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4253, 4194301, F4, 2, 28) (dual of [(4194301, 2), 8388349, 29]-NRT-code), using
- strength reduction [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
- strength reduction [i] based on linear OOA(4253, 4194301, F4, 2, 29) (dual of [(4194301, 2), 8388349, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4253, 4194301, F4, 2, 28) (dual of [(4194301, 2), 8388349, 29]-NRT-code), using
(253−28, 253, large)-Net in Base 4 — Upper bound on s
There is no (225, 253, large)-net in base 4, because
- 26 times m-reduction [i] would yield (225, 227, large)-net in base 4, but