Best Known (249−45, 249, s)-Nets in Base 4
(249−45, 249, 1560)-Net over F4 — Constructive and digital
Digital (204, 249, 1560)-net over F4, using
- 41 times duplication [i] based on digital (203, 248, 1560)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 29, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- 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
- digital (7, 29, 21)-net over F4, using
- (u, u+v)-construction [i] based on
(249−45, 249, 16445)-Net over F4 — Digital
Digital (204, 249, 16445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4249, 16445, F4, 45) (dual of [16445, 16196, 46]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4246, 16440, F4, 45) (dual of [16440, 16194, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(36) [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(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(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(44) ⊂ Ce(36) [i] based on
- linear OA(4246, 16442, F4, 43) (dual of [16442, 16196, 44]-code), using Gilbert–Varšamov bound and bm = 4246 > Vbs−1(k−1) = 86 697833 185116 284300 590827 640063 084176 888237 431648 913177 580174 273571 403649 739246 386107 088928 384677 334662 587947 817038 965171 887227 218438 106111 511952 [i]
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4246, 16440, F4, 45) (dual of [16440, 16194, 46]-code), using
- construction X with Varšamov bound [i] based on
(249−45, 249, large)-Net in Base 4 — Upper bound on s
There is no (204, 249, large)-net in base 4, because
- 43 times m-reduction [i] would yield (204, 206, large)-net in base 4, but