Best Known (252−47, 252, s)-Nets in Base 4
(252−47, 252, 1554)-Net over F4 — Constructive and digital
Digital (205, 252, 1554)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 27, 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 (178, 225, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
- digital (4, 27, 15)-net over F4, using
(252−47, 252, 13367)-Net over F4 — Digital
Digital (205, 252, 13367)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4252, 13367, F4, 47) (dual of [13367, 13115, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4252, 16412, F4, 47) (dual of [16412, 16160, 48]-code), using
- construction XX applied to Ce(46) ⊂ Ce(42) ⊂ Ce(41) [i] based on
- linear OA(4246, 16384, F4, 47) (dual of [16384, 16138, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(4225, 16384, F4, 43) (dual of [16384, 16159, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(46) ⊂ Ce(42) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(4252, 16412, F4, 47) (dual of [16412, 16160, 48]-code), using
(252−47, 252, large)-Net in Base 4 — Upper bound on s
There is no (205, 252, large)-net in base 4, because
- 45 times m-reduction [i] would yield (205, 207, large)-net in base 4, but