Best Known (241−43, 241, s)-Nets in Base 4
(241−43, 241, 1560)-Net over F4 — Constructive and digital
Digital (198, 241, 1560)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 28, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (170, 213, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 71, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 71, 513)-net over F64, using
- digital (7, 28, 21)-net over F4, using
(241−43, 241, 16445)-Net over F4 — Digital
Digital (198, 241, 16445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4241, 16445, F4, 43) (dual of [16445, 16204, 44]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4239, 16441, F4, 43) (dual of [16441, 16202, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,17]) [i] based on
- linear OA(4225, 16385, F4, 43) (dual of [16385, 16160, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(4183, 16385, F4, 35) (dual of [16385, 16202, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to C([0,21]) ⊂ C([0,17]) [i] based on
- linear OA(4239, 16443, F4, 42) (dual of [16443, 16204, 43]-code), using Gilbert–Varšamov bound and bm = 4239 > Vbs−1(k−1) = 74193 843176 361427 585016 198057 428502 818826 302787 739265 300786 960006 686181 397270 882758 417457 205843 077620 432304 126871 209088 726628 945073 027999 946655 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4239, 16441, F4, 43) (dual of [16441, 16202, 44]-code), using
- construction X with Varšamov bound [i] based on
(241−43, 241, large)-Net in Base 4 — Upper bound on s
There is no (198, 241, large)-net in base 4, because
- 41 times m-reduction [i] would yield (198, 200, large)-net in base 4, but