Best Known (228−43, 228, s)-Nets in Base 4
(228−43, 228, 1539)-Net over F4 — Constructive and digital
Digital (185, 228, 1539)-net over F4, using
- t-expansion [i] based on digital (184, 228, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (184, 234, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 78, 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, 78, 513)-net over F64, using
- 6 times m-reduction [i] based on digital (184, 234, 1539)-net over F4, using
(228−43, 228, 11559)-Net over F4 — Digital
Digital (185, 228, 11559)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4228, 11559, F4, 43) (dual of [11559, 11331, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4228, 16402, F4, 43) (dual of [16402, 16174, 44]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4226, 16400, F4, 43) (dual of [16400, 16174, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,20]) [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(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(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4226, 16400, F4, 43) (dual of [16400, 16174, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4228, 16402, F4, 43) (dual of [16402, 16174, 44]-code), using
(228−43, 228, large)-Net in Base 4 — Upper bound on s
There is no (185, 228, large)-net in base 4, because
- 41 times m-reduction [i] would yield (185, 187, large)-net in base 4, but