Best Known (198, 198+44, s)-Nets in Base 4
(198, 198+44, 1554)-Net over F4 — Constructive and digital
Digital (198, 242, 1554)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 26, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (172, 216, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 72, 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, 72, 513)-net over F64, using
- digital (4, 26, 15)-net over F4, using
(198, 198+44, 15647)-Net over F4 — Digital
Digital (198, 242, 15647)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4242, 15647, F4, 44) (dual of [15647, 15405, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4242, 16406, F4, 44) (dual of [16406, 16164, 45]-code), using
- construction X applied to C([0,22]) ⊂ C([0,20]) [i] based on
- linear OA(4239, 16385, F4, 45) (dual of [16385, 16146, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (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(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to C([0,22]) ⊂ C([0,20]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4242, 16406, F4, 44) (dual of [16406, 16164, 45]-code), using
(198, 198+44, large)-Net in Base 4 — Upper bound on s
There is no (198, 242, large)-net in base 4, because
- 42 times m-reduction [i] would yield (198, 200, large)-net in base 4, but