Best Known (215−40, 215, s)-Nets in Base 4
(215−40, 215, 1539)-Net over F4 — Constructive and digital
Digital (175, 215, 1539)-net over F4, using
- t-expansion [i] based on digital (174, 215, 1539)-net over F4, using
- 4 times m-reduction [i] based on digital (174, 219, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 73, 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, 73, 513)-net over F64, using
- 4 times m-reduction [i] based on digital (174, 219, 1539)-net over F4, using
(215−40, 215, 12279)-Net over F4 — Digital
Digital (175, 215, 12279)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4215, 12279, F4, 40) (dual of [12279, 12064, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4215, 16409, F4, 40) (dual of [16409, 16194, 41]-code), using
- 1 times truncation [i] based on linear OA(4216, 16410, F4, 41) (dual of [16410, 16194, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(40) ⊂ Ce(36) [i] based on
- 1 times truncation [i] based on linear OA(4216, 16410, F4, 41) (dual of [16410, 16194, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4215, 16409, F4, 40) (dual of [16409, 16194, 41]-code), using
(215−40, 215, 8209732)-Net in Base 4 — Upper bound on s
There is no (175, 215, 8209733)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2772 670380 625475 799800 314617 311274 292966 861715 278703 064224 396913 471561 231018 908718 373635 645118 040820 129345 039414 247382 794472 797564 > 4215 [i]