Best Known (220−39, 220, s)-Nets in Base 4
(220−39, 220, 1544)-Net over F4 — Constructive and digital
Digital (181, 220, 1544)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 19, 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 (162, 201, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 67, 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, 67, 513)-net over F64, using
- digital (0, 19, 5)-net over F4, using
(220−39, 220, 16444)-Net over F4 — Digital
Digital (181, 220, 16444)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4220, 16444, F4, 39) (dual of [16444, 16224, 40]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4218, 16440, F4, 39) (dual of [16440, 16222, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(30) [i] based on
- 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(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(38) ⊂ Ce(30) [i] based on
- linear OA(4218, 16442, F4, 38) (dual of [16442, 16224, 39]-code), using Gilbert–Varšamov bound and bm = 4218 > Vbs−1(k−1) = 30677 150176 584088 669507 683963 646940 620101 150064 912773 882772 440054 274121 533020 149443 758716 347565 879824 992530 439843 307047 728117 595244 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4218, 16440, F4, 39) (dual of [16440, 16222, 40]-code), using
- construction X with Varšamov bound [i] based on
(220−39, 220, large)-Net in Base 4 — Upper bound on s
There is no (181, 220, large)-net in base 4, because
- 37 times m-reduction [i] would yield (181, 183, large)-net in base 4, but