Best Known (238−44, 238, s)-Nets in Base 4
(238−44, 238, 1544)-Net over F4 — Constructive and digital
Digital (194, 238, 1544)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 22, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (172, 216, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 72, 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, 72, 513)-net over F64, using
- digital (0, 22, 5)-net over F4, using
(238−44, 238, 13707)-Net over F4 — Digital
Digital (194, 238, 13707)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4238, 13707, F4, 44) (dual of [13707, 13469, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4238, 16412, F4, 44) (dual of [16412, 16174, 45]-code), using
- 1 times truncation [i] based on linear OA(4239, 16413, F4, 45) (dual of [16413, 16174, 46]-code), using
- construction XX applied to Ce(44) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 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(4204, 16384, F4, 39) (dual of [16384, 16180, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [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(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(44) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- 1 times truncation [i] based on linear OA(4239, 16413, F4, 45) (dual of [16413, 16174, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4238, 16412, F4, 44) (dual of [16412, 16174, 45]-code), using
(238−44, 238, large)-Net in Base 4 — Upper bound on s
There is no (194, 238, large)-net in base 4, because
- 42 times m-reduction [i] would yield (194, 196, large)-net in base 4, but