Best Known (170, 195, s)-Nets in Base 4
(170, 195, 87391)-Net over F4 — Constructive and digital
Digital (170, 195, 87391)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-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 (2, 14, 10)-net over F4, using
(170, 195, 524321)-Net over F4 — Digital
Digital (170, 195, 524321)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4195, 524321, F4, 2, 25) (dual of [(524321, 2), 1048447, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4195, 1048642, F4, 25) (dual of [1048642, 1048447, 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(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(4195, 1048642, F4, 25) (dual of [1048642, 1048447, 26]-code), using
(170, 195, large)-Net in Base 4 — Upper bound on s
There is no (170, 195, large)-net in base 4, because
- 23 times m-reduction [i] would yield (170, 172, large)-net in base 4, but