Best Known (242−17, 242, s)-Nets in Base 4
(242−17, 242, 4198401)-Net over F4 — Constructive and digital
Digital (225, 242, 4198401)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (38, 46, 4101)-net over F4, using
- net defined by OOA [i] based on linear OOA(446, 4101, F4, 8, 8) (dual of [(4101, 8), 32762, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(446, 16404, F4, 8) (dual of [16404, 16358, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(446, 16405, F4, 8) (dual of [16405, 16359, 9]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(446, 16405, F4, 8) (dual of [16405, 16359, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(446, 16404, F4, 8) (dual of [16404, 16358, 9]-code), using
- net defined by OOA [i] based on linear OOA(446, 4101, F4, 8, 8) (dual of [(4101, 8), 32762, 9]-NRT-code), using
- digital (179, 196, 4194300)-net over F4, using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- digital (38, 46, 4101)-net over F4, using
(242−17, 242, large)-Net over F4 — Digital
Digital (225, 242, large)-net over F4, using
- t-expansion [i] based on digital (222, 242, large)-net over F4, using
- 5 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
- 5 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
(242−17, 242, large)-Net in Base 4 — Upper bound on s
There is no (225, 242, large)-net in base 4, because
- 15 times m-reduction [i] would yield (225, 227, large)-net in base 4, but