Best Known (223, 223+21, s)-Nets in Base 4
(223, 223+21, 3355440)-Net over F4 — Constructive and digital
Digital (223, 244, 3355440)-net over F4, using
- trace code for nets [i] based on digital (101, 122, 1677720)-net over F16, using
- net defined by OOA [i] based on linear OOA(16122, 1677720, F16, 22, 21) (dual of [(1677720, 22), 36909718, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16122, 8388601, F16, 2, 21) (dual of [(8388601, 2), 16777080, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16122, 8388602, F16, 2, 21) (dual of [(8388602, 2), 16777082, 22]-NRT-code), using
- trace code [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- OOA 2-folding [i] based on linear OA(25661, 8388602, F256, 21) (dual of [8388602, 8388541, 22]-code), using
- trace code [i] based on linear OOA(25661, 4194301, F256, 2, 21) (dual of [(4194301, 2), 8388541, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16122, 8388602, F16, 2, 21) (dual of [(8388602, 2), 16777082, 22]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16122, 8388601, F16, 2, 21) (dual of [(8388601, 2), 16777080, 22]-NRT-code), using
- net defined by OOA [i] based on linear OOA(16122, 1677720, F16, 22, 21) (dual of [(1677720, 22), 36909718, 22]-NRT-code), using
(223, 223+21, large)-Net over F4 — Digital
Digital (223, 244, large)-net over F4, using
- t-expansion [i] based on digital (222, 244, large)-net over F4, using
- 3 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 3 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
(223, 223+21, large)-Net in Base 4 — Upper bound on s
There is no (223, 244, large)-net in base 4, because
- 19 times m-reduction [i] would yield (223, 225, large)-net in base 4, but