Best Known (181, 222, s)-Nets in Base 4
(181, 222, 1539)-Net over F4 — Constructive and digital
Digital (181, 222, 1539)-net over F4, using
- t-expansion [i] based on digital (180, 222, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (180, 228, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 76, 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, 76, 513)-net over F64, using
- 6 times m-reduction [i] based on digital (180, 228, 1539)-net over F4, using
(181, 222, 13211)-Net over F4 — Digital
Digital (181, 222, 13211)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4222, 13211, F4, 41) (dual of [13211, 12989, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 16424, F4, 41) (dual of [16424, 16202, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4221, 16423, F4, 41) (dual of [16423, 16202, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [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(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(410, 38, F4, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4221, 16423, F4, 41) (dual of [16423, 16202, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 16424, F4, 41) (dual of [16424, 16202, 42]-code), using
(181, 222, large)-Net in Base 4 — Upper bound on s
There is no (181, 222, large)-net in base 4, because
- 39 times m-reduction [i] would yield (181, 183, large)-net in base 4, but