Best Known (172, 172+25, s)-Nets in Base 4
(172, 172+25, 87396)-Net over F4 — Constructive and digital
Digital (172, 197, 87396)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 16, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-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 (4, 16, 15)-net over F4, using
(172, 172+25, 524326)-Net over F4 — Digital
Digital (172, 197, 524326)-net over F4, using
- 41 times duplication [i] based on digital (171, 196, 524326)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4196, 524326, F4, 2, 25) (dual of [(524326, 2), 1048456, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4196, 1048652, F4, 25) (dual of [1048652, 1048456, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [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(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(415, 75, F4, 7) (dual of [75, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(415, 85, F4, 7) (dual of [85, 70, 8]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(4196, 1048652, F4, 25) (dual of [1048652, 1048456, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4196, 524326, F4, 2, 25) (dual of [(524326, 2), 1048456, 26]-NRT-code), using
(172, 172+25, large)-Net in Base 4 — Upper bound on s
There is no (172, 197, large)-net in base 4, because
- 23 times m-reduction [i] would yield (172, 174, large)-net in base 4, but