Best Known (177, 177+17, s)-Nets in Base 4
(177, 177+17, 2097167)-Net over F4 — Constructive and digital
Digital (177, 194, 2097167)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 17)-net over F4, using
- digital (165, 182, 2097150)-net over F4, using
- net defined by OOA [i] based on linear OOA(4182, 2097150, F4, 18, 17) (dual of [(2097150, 18), 37748518, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(4182, 8388601, F4, 2, 17) (dual of [(8388601, 2), 16777020, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4182, 8388602, F4, 2, 17) (dual of [(8388602, 2), 16777022, 18]-NRT-code), using
- trace code [i] based on linear OOA(1691, 4194301, F16, 2, 17) (dual of [(4194301, 2), 8388511, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1691, 8388602, F16, 17) (dual of [8388602, 8388511, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1691, large, F16, 17) (dual of [large, large−91, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(1691, large, F16, 17) (dual of [large, large−91, 18]-code), using
- OOA 2-folding [i] based on linear OA(1691, 8388602, F16, 17) (dual of [8388602, 8388511, 18]-code), using
- trace code [i] based on linear OOA(1691, 4194301, F16, 2, 17) (dual of [(4194301, 2), 8388511, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4182, 8388602, F4, 2, 17) (dual of [(8388602, 2), 16777022, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(4182, 8388601, F4, 2, 17) (dual of [(8388601, 2), 16777020, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4182, 2097150, F4, 18, 17) (dual of [(2097150, 18), 37748518, 18]-NRT-code), using
(177, 177+17, large)-Net over F4 — Digital
Digital (177, 194, large)-net over F4, using
- 3 times m-reduction [i] based on digital (177, 197, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4197, large, F4, 20) (dual of [large, large−197, 21]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 17 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4197, large, F4, 20) (dual of [large, large−197, 21]-code), using
(177, 177+17, large)-Net in Base 4 — Upper bound on s
There is no (177, 194, large)-net in base 4, because
- 15 times m-reduction [i] would yield (177, 179, large)-net in base 4, but