Best Known (164, 199, s)-Nets in Base 4
(164, 199, 1539)-Net over F4 — Constructive and digital
Digital (164, 199, 1539)-net over F4, using
- 5 times m-reduction [i] based on digital (164, 204, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 68, 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, 68, 513)-net over F64, using
(164, 199, 16445)-Net over F4 — Digital
Digital (164, 199, 16445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4199, 16445, F4, 35) (dual of [16445, 16246, 36]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4197, 16441, F4, 35) (dual of [16441, 16244, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- 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(4141, 16385, F4, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(4197, 16443, F4, 34) (dual of [16443, 16246, 35]-code), using Gilbert–Varšamov bound and bm = 4197 > Vbs−1(k−1) = 8301 460968 174328 472586 192372 018618 735699 236278 091268 207983 224466 311168 108113 147307 121069 062868 630606 144670 217355 190709 [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(4197, 16441, F4, 35) (dual of [16441, 16244, 36]-code), using
- construction X with Varšamov bound [i] based on
(164, 199, large)-Net in Base 4 — Upper bound on s
There is no (164, 199, large)-net in base 4, because
- 33 times m-reduction [i] would yield (164, 166, large)-net in base 4, but