Best Known (168, 168+25, s)-Nets in Base 4
(168, 168+25, 87386)-Net over F4 — Constructive and digital
Digital (168, 193, 87386)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (156, 181, 87381)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 87381, F4, 25, 25) (dual of [(87381, 25), 2184344, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4181, 1048573, F4, 25) (dual of [1048573, 1048392, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4181, 1048573, F4, 25) (dual of [1048573, 1048392, 26]-code), using
- net defined by OOA [i] based on linear OOA(4181, 87381, F4, 25, 25) (dual of [(87381, 25), 2184344, 26]-NRT-code), using
- digital (0, 12, 5)-net over F4, using
(168, 168+25, 508970)-Net over F4 — Digital
Digital (168, 193, 508970)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4193, 508970, F4, 2, 25) (dual of [(508970, 2), 1017747, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4193, 524314, F4, 2, 25) (dual of [(524314, 2), 1048435, 26]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4192, 524314, F4, 2, 25) (dual of [(524314, 2), 1048436, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4192, 1048628, F4, 25) (dual of [1048628, 1048436, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4191, 1048627, F4, 25) (dual of [1048627, 1048436, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4191, 1048627, F4, 25) (dual of [1048627, 1048436, 26]-code), using
- OOA 2-folding [i] based on linear OA(4192, 1048628, F4, 25) (dual of [1048628, 1048436, 26]-code), using
- 41 times duplication [i] based on linear OOA(4192, 524314, F4, 2, 25) (dual of [(524314, 2), 1048436, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4193, 524314, F4, 2, 25) (dual of [(524314, 2), 1048435, 26]-NRT-code), using
(168, 168+25, large)-Net in Base 4 — Upper bound on s
There is no (168, 193, large)-net in base 4, because
- 23 times m-reduction [i] would yield (168, 170, large)-net in base 4, but