Best Known (117−17, 117, s)-Nets in Base 4
(117−17, 117, 32773)-Net over F4 — Constructive and digital
Digital (100, 117, 32773)-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 (92, 109, 32768)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 32768, F4, 17, 17) (dual of [(32768, 17), 556947, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, 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(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using
- net defined by OOA [i] based on linear OOA(4109, 32768, F4, 17, 17) (dual of [(32768, 17), 556947, 18]-NRT-code), using
- digital (0, 8, 5)-net over F4, using
(117−17, 117, 131090)-Net over F4 — Digital
Digital (100, 117, 131090)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4117, 131090, F4, 2, 17) (dual of [(131090, 2), 262063, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4117, 262180, F4, 17) (dual of [262180, 262063, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4114, 262176, F4, 17) (dual of [262176, 262062, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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
- linear OA(4114, 262177, F4, 14) (dual of [262177, 262063, 15]-code), using Gilbert–Varšamov bound and bm = 4114 > Vbs−1(k−1) = 7 077536 142071 768841 898351 771288 368189 050973 506714 868838 032300 649069 [i]
- linear OA(42, 3, F4, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,4)), using
- dual of repetition code with length 3 [i]
- linear OA(4114, 262176, F4, 17) (dual of [262176, 262062, 18]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(4117, 262180, F4, 17) (dual of [262180, 262063, 18]-code), using
(117−17, 117, large)-Net in Base 4 — Upper bound on s
There is no (100, 117, large)-net in base 4, because
- 15 times m-reduction [i] would yield (100, 102, large)-net in base 4, but