Best Known (231, 231+21, s)-Nets in Base 4
(231, 231+21, 3355440)-Net over F4 — Constructive and digital
Digital (231, 252, 3355440)-net over F4, using
- 48 times duplication [i] based on 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
- trace code for nets [i] based on digital (101, 122, 1677720)-net over F16, using
(231, 231+21, large)-Net over F4 — Digital
Digital (231, 252, large)-net over F4, using
- 5 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
(231, 231+21, large)-Net in Base 4 — Upper bound on s
There is no (231, 252, large)-net in base 4, because
- 19 times m-reduction [i] would yield (231, 233, large)-net in base 4, but