Best Known (149, 149+13, s)-Nets in Base 4
(149, 149+13, 5592448)-Net over F4 — Constructive and digital
Digital (149, 162, 5592448)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (8, 14, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 7, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 7, 24)-net over F16, using
- digital (135, 148, 5592400)-net over F4, using
- trace code for nets [i] based on digital (61, 74, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1674, 2796200, F16, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1674, 8388601, F16, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1674, 8388602, F16, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
- trace code [i] based on linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- trace code [i] based on linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1674, 8388602, F16, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1674, 8388601, F16, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1674, 2796200, F16, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
- trace code for nets [i] based on digital (61, 74, 2796200)-net over F16, using
- digital (8, 14, 48)-net over F4, using
(149, 149+13, large)-Net over F4 — Digital
Digital (149, 162, large)-net over F4, using
- 4 times m-reduction [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
(149, 149+13, large)-Net in Base 4 — Upper bound on s
There is no (149, 162, large)-net in base 4, because
- 11 times m-reduction [i] would yield (149, 151, large)-net in base 4, but