Best Known (256−35, 256, s)-Nets in Base 4
(256−35, 256, 15435)-Net over F4 — Constructive and digital
Digital (221, 256, 15435)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 21, 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 (200, 235, 15420)-net over F4, using
- net defined by OOA [i] based on linear OOA(4235, 15420, F4, 35, 35) (dual of [(15420, 35), 539465, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4235, 262141, F4, 35) (dual of [262141, 261906, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- discarding factors / shortening the dual code based on linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4235, 262141, F4, 35) (dual of [262141, 261906, 36]-code), using
- net defined by OOA [i] based on linear OOA(4235, 15420, F4, 35, 35) (dual of [(15420, 35), 539465, 36]-NRT-code), using
- digital (4, 21, 15)-net over F4, using
(256−35, 256, 196995)-Net over F4 — Digital
Digital (221, 256, 196995)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4256, 196995, F4, 35) (dual of [196995, 196739, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4256, 262168, F4, 35) (dual of [262168, 261912, 36]-code), using
- (u, u+v)-construction [i] based on
- linear OA(421, 23, F4, 17) (dual of [23, 2, 18]-code), using
- 2 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- repeating each code word 5 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- extended Reed–Solomon code RSe(2,4) [i]
- Simplex code S(2,4) [i]
- repeating each code word 5 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- 2 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- linear OA(4235, 262145, F4, 35) (dual of [262145, 261910, 36]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(421, 23, F4, 17) (dual of [23, 2, 18]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4256, 262168, F4, 35) (dual of [262168, 261912, 36]-code), using
(256−35, 256, large)-Net in Base 4 — Upper bound on s
There is no (221, 256, large)-net in base 4, because
- 33 times m-reduction [i] would yield (221, 223, large)-net in base 4, but