Best Known (157, 157+12, s)-Nets in Base 4
(157, 157+12, 5614248)-Net over F4 — Constructive and digital
Digital (157, 169, 5614248)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (27, 33, 21848)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 21848, F4, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(433, 65544, F4, 6) (dual of [65544, 65511, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(433, 65544, F4, 6) (dual of [65544, 65511, 7]-code), using
- net defined by OOA [i] based on linear OOA(433, 21848, F4, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
- digital (124, 136, 5592400)-net over F4, using
- trace code for nets [i] based on digital (56, 68, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- trace code for nets [i] based on digital (56, 68, 2796200)-net over F16, using
- digital (27, 33, 21848)-net over F4, using
(157, 157+12, large)-Net over F4 — Digital
Digital (157, 169, large)-net over F4, using
- 43 times duplication [i] based on digital (154, 166, large)-net over F4, using
- t-expansion [i] based on digital (149, 166, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4166, large, F4, 17) (dual of [large, large−166, 18]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 21 times code embedding in larger space [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4166, large, F4, 17) (dual of [large, large−166, 18]-code), using
- t-expansion [i] based on digital (149, 166, large)-net over F4, using
(157, 157+12, large)-Net in Base 4 — Upper bound on s
There is no (157, 169, large)-net in base 4, because
- 10 times m-reduction [i] would yield (157, 159, large)-net in base 4, but