Best Known (195, 195+45, s)-Nets in Base 4
(195, 195+45, 1539)-Net over F4 — Constructive and digital
Digital (195, 240, 1539)-net over F4, using
- t-expansion [i] based on digital (194, 240, 1539)-net over F4, using
- 9 times m-reduction [i] based on digital (194, 249, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 83, 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, 83, 513)-net over F64, using
- 9 times m-reduction [i] based on digital (194, 249, 1539)-net over F4, using
(195, 195+45, 12459)-Net over F4 — Digital
Digital (195, 240, 12459)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4240, 12459, F4, 45) (dual of [12459, 12219, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 16400, F4, 45) (dual of [16400, 16160, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,21]) [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(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(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([0,22]) ⊂ C([0,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4240, 16400, F4, 45) (dual of [16400, 16160, 46]-code), using
(195, 195+45, large)-Net in Base 4 — Upper bound on s
There is no (195, 240, large)-net in base 4, because
- 43 times m-reduction [i] would yield (195, 197, large)-net in base 4, but