Best Known (185, 185+40, s)-Nets in Base 4
(185, 185+40, 1548)-Net over F4 — Constructive and digital
Digital (185, 225, 1548)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 21, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- 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
- digital (1, 21, 9)-net over F4, using
(185, 185+40, 16442)-Net over F4 — Digital
Digital (185, 225, 16442)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4225, 16442, F4, 40) (dual of [16442, 16217, 41]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4224, 16440, F4, 40) (dual of [16440, 16216, 41]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(4211, 16385, F4, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(4224, 16441, F4, 39) (dual of [16441, 16217, 40]-code), using Gilbert–Varšamov bound and bm = 4224 > Vbs−1(k−1) = 39 637509 447789 024412 137436 951917 161835 935726 781549 691003 839799 715540 415922 004542 578898 489367 429927 129172 658550 707372 492455 369864 048061 [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(4224, 16440, F4, 40) (dual of [16440, 16216, 41]-code), using
- construction X with Varšamov bound [i] based on
(185, 185+40, large)-Net in Base 4 — Upper bound on s
There is no (185, 225, large)-net in base 4, because
- 38 times m-reduction [i] would yield (185, 187, large)-net in base 4, but