Best Known (112, 112+17, s)-Nets in Base 4
(112, 112+17, 131077)-Net over F4 — Constructive and digital
Digital (112, 129, 131077)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (104, 121, 131072)-net over F4, using
- net defined by OOA [i] based on linear OOA(4121, 131072, F4, 17, 17) (dual of [(131072, 17), 2228103, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on 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]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using
- net defined by OOA [i] based on linear OOA(4121, 131072, F4, 17, 17) (dual of [(131072, 17), 2228103, 18]-NRT-code), using
- digital (0, 8, 5)-net over F4, using
(112, 112+17, 524307)-Net over F4 — Digital
Digital (112, 129, 524307)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4129, 524307, F4, 2, 17) (dual of [(524307, 2), 1048485, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4127, 524306, F4, 2, 17) (dual of [(524306, 2), 1048485, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4127, 1048612, F4, 17) (dual of [1048612, 1048485, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4126, 1048611, F4, 17) (dual of [1048611, 1048485, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4126, 1048611, F4, 17) (dual of [1048611, 1048485, 18]-code), using
- OOA 2-folding [i] based on linear OA(4127, 1048612, F4, 17) (dual of [1048612, 1048485, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4127, 524306, F4, 2, 17) (dual of [(524306, 2), 1048485, 18]-NRT-code), using
(112, 112+17, large)-Net in Base 4 — Upper bound on s
There is no (112, 129, large)-net in base 4, because
- 15 times m-reduction [i] would yield (112, 114, large)-net in base 4, but