Best Known (177, 177+39, s)-Nets in Base 4
(177, 177+39, 1539)-Net over F4 — Constructive and digital
Digital (177, 216, 1539)-net over F4, using
- t-expansion [i] based on digital (176, 216, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (176, 222, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 74, 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, 74, 513)-net over F64, using
- 6 times m-reduction [i] based on digital (176, 222, 1539)-net over F4, using
(177, 177+39, 15361)-Net over F4 — Digital
Digital (177, 216, 15361)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4216, 15361, F4, 39) (dual of [15361, 15145, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 16418, F4, 39) (dual of [16418, 16202, 40]-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(45, 33, F4, 3) (dual of [33, 28, 4]-code or 33-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4216, 16418, F4, 39) (dual of [16418, 16202, 40]-code), using
(177, 177+39, large)-Net in Base 4 — Upper bound on s
There is no (177, 216, large)-net in base 4, because
- 37 times m-reduction [i] would yield (177, 179, large)-net in base 4, but