Best Known (257−20, 257, s)-Nets in Base 4
(257−20, 257, 3355506)-Net over F4 — Constructive and digital
Digital (237, 257, 3355506)-net over F4, using
- 41 times duplication [i] based on digital (236, 256, 3355506)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (14, 24, 66)-net over F4, using
- trace code for nets [i] based on digital (2, 12, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- trace code for nets [i] based on digital (2, 12, 33)-net over F16, using
- digital (212, 232, 3355440)-net over F4, using
- trace code for nets [i] based on digital (96, 116, 1677720)-net over F16, using
- net defined by OOA [i] based on linear OOA(16116, 1677720, F16, 22, 20) (dual of [(1677720, 22), 36909724, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16116, 8388601, F16, 2, 20) (dual of [(8388601, 2), 16777086, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16116, 8388602, F16, 2, 20) (dual of [(8388602, 2), 16777088, 21]-NRT-code), using
- trace code [i] based on linear OOA(25658, 4194301, F256, 2, 20) (dual of [(4194301, 2), 8388544, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25658, 8388602, F256, 20) (dual of [8388602, 8388544, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- OOA 2-folding [i] based on linear OA(25658, 8388602, F256, 20) (dual of [8388602, 8388544, 21]-code), using
- trace code [i] based on linear OOA(25658, 4194301, F256, 2, 20) (dual of [(4194301, 2), 8388544, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16116, 8388602, F16, 2, 20) (dual of [(8388602, 2), 16777088, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(16116, 8388601, F16, 2, 20) (dual of [(8388601, 2), 16777086, 21]-NRT-code), using
- net defined by OOA [i] based on linear OOA(16116, 1677720, F16, 22, 20) (dual of [(1677720, 22), 36909724, 21]-NRT-code), using
- trace code for nets [i] based on digital (96, 116, 1677720)-net over F16, using
- digital (14, 24, 66)-net over F4, using
- (u, u+v)-construction [i] based on
(257−20, 257, large)-Net over F4 — Digital
Digital (237, 257, large)-net over F4, using
- t-expansion [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
(257−20, 257, large)-Net in Base 4 — Upper bound on s
There is no (237, 257, large)-net in base 4, because
- 18 times m-reduction [i] would yield (237, 239, large)-net in base 4, but