Best Known (250−45, 250, s)-Nets in Base 4
(250−45, 250, 1561)-Net over F4 — Constructive and digital
Digital (205, 250, 1561)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 31, 22)-net over F4, using
- net from sequence [i] based on digital (9, 21)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 9 and N(F) ≥ 22, using
- net from sequence [i] based on digital (9, 21)-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 (9, 31, 22)-net over F4, using
(250−45, 250, 16447)-Net over F4 — Digital
Digital (205, 250, 16447)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4250, 16447, F4, 45) (dual of [16447, 16197, 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, 16443, F4, 43) (dual of [16443, 16197, 44]-code), using Gilbert–Varšamov bound and bm = 4246 > Vbs−1(k−1) = 86 919859 693534 699858 603338 551172 944189 456016 780443 047038 723555 346524 679859 857964 165522 152943 794079 070793 308453 658832 487562 688969 010691 787884 880008 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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
(250−45, 250, large)-Net in Base 4 — Upper bound on s
There is no (205, 250, large)-net in base 4, because
- 43 times m-reduction [i] would yield (205, 207, large)-net in base 4, but