Best Known (226−20, 226, s)-Nets in Base 4
(226−20, 226, 1677720)-Net over F4 — Constructive and digital
Digital (206, 226, 1677720)-net over F4, using
- 48 times duplication [i] based on digital (198, 218, 1677720)-net over F4, using
- net defined by OOA [i] based on linear OOA(4218, 1677720, F4, 22, 20) (dual of [(1677720, 22), 36909622, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(4218, 8388601, F4, 2, 20) (dual of [(8388601, 2), 16776984, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4218, 8388602, F4, 2, 20) (dual of [(8388602, 2), 16776986, 21]-NRT-code), using
- trace code [i] based on linear OOA(16109, 4194301, F16, 2, 20) (dual of [(4194301, 2), 8388493, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(16109, 8388602, F16, 20) (dual of [8388602, 8388493, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- OOA 2-folding [i] based on linear OA(16109, 8388602, F16, 20) (dual of [8388602, 8388493, 21]-code), using
- trace code [i] based on linear OOA(16109, 4194301, F16, 2, 20) (dual of [(4194301, 2), 8388493, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4218, 8388602, F4, 2, 20) (dual of [(8388602, 2), 16776986, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(4218, 8388601, F4, 2, 20) (dual of [(8388601, 2), 16776984, 21]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4218, 1677720, F4, 22, 20) (dual of [(1677720, 22), 36909622, 21]-NRT-code), using
(226−20, 226, large)-Net over F4 — Digital
Digital (206, 226, large)-net over F4, using
- t-expansion [i] based on digital (204, 226, large)-net over F4, using
- 1 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 1 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
(226−20, 226, large)-Net in Base 4 — Upper bound on s
There is no (206, 226, large)-net in base 4, because
- 18 times m-reduction [i] would yield (206, 208, large)-net in base 4, but