Best Known (164, 202, s)-Nets in Base 4
(164, 202, 1539)-Net over F4 — Constructive and digital
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
(164, 202, 10914)-Net over F4 — Digital
Digital (164, 202, 10914)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4202, 10914, F4, 38) (dual of [10914, 10712, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4202, 16410, F4, 38) (dual of [16410, 16208, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(4197, 16384, F4, 38) (dual of [16384, 16187, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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 Ce(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4202, 16410, F4, 38) (dual of [16410, 16208, 39]-code), using
(164, 202, 6651836)-Net in Base 4 — Upper bound on s
There is no (164, 202, 6651837)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 41 316032 558542 605939 348683 680092 560499 405741 332910 478383 715370 274979 846770 843295 765530 270171 107300 081686 736040 627156 671740 > 4202 [i]