Best Known (165, 165+37, s)-Nets in Base 4
(165, 165+37, 1539)-Net over F4 — Constructive and digital
Digital (165, 202, 1539)-net over F4, using
- t-expansion [i] based on digital (164, 202, 1539)-net over F4, using
- 2 times m-reduction [i] based on digital (164, 204, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 68, 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, 68, 513)-net over F64, using
- 2 times m-reduction [i] based on digital (164, 204, 1539)-net over F4, using
(165, 165+37, 13267)-Net over F4 — Digital
Digital (165, 202, 13267)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4202, 13267, F4, 37) (dual of [13267, 13065, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4202, 16418, F4, 37) (dual of [16418, 16216, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(4197, 16385, F4, 37) (dual of [16385, 16188, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4202, 16418, F4, 37) (dual of [16418, 16216, 38]-code), using
(165, 165+37, large)-Net in Base 4 — Upper bound on s
There is no (165, 202, large)-net in base 4, because
- 35 times m-reduction [i] would yield (165, 167, large)-net in base 4, but