Best Known (197, 240, s)-Nets in Base 4
(197, 240, 1556)-Net over F4 — Constructive and digital
Digital (197, 240, 1556)-net over F4, using
- 41 times duplication [i] based on digital (196, 239, 1556)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 26, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-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 (5, 26, 17)-net over F4, using
- (u, u+v)-construction [i] based on
(197, 240, 16443)-Net over F4 — Digital
Digital (197, 240, 16443)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4240, 16443, F4, 43) (dual of [16443, 16203, 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, 16442, F4, 42) (dual of [16442, 16203, 43]-code), using Gilbert–Varšamov bound and bm = 4239 > Vbs−1(k−1) = 74008 836139 471852 670836 970369 953337 522593 116264 711287 047793 690984 425403 372905 926471 688005 136467 245376 906835 280597 840796 933913 930751 227257 789352 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
(197, 240, large)-Net in Base 4 — Upper bound on s
There is no (197, 240, large)-net in base 4, because
- 41 times m-reduction [i] would yield (197, 199, large)-net in base 4, but