Best Known (165−29, 165, s)-Nets in Base 4
(165−29, 165, 1272)-Net over F4 — Constructive and digital
Digital (136, 165, 1272)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (31, 45, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- digital (91, 120, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 30, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 30, 258)-net over F256, using
- digital (31, 45, 240)-net over F4, using
(165−29, 165, 16445)-Net over F4 — Digital
Digital (136, 165, 16445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4165, 16445, F4, 29) (dual of [16445, 16280, 30]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4162, 16440, F4, 29) (dual of [16440, 16278, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(28) ⊂ Ce(20) [i] based on
- linear OA(4162, 16442, F4, 27) (dual of [16442, 16280, 28]-code), using Gilbert–Varšamov bound and bm = 4162 > Vbs−1(k−1) = 254277 229678 767333 724530 915378 760991 998755 967322 095445 840310 441021 103141 720896 948748 295649 426400 [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(4162, 16440, F4, 29) (dual of [16440, 16278, 30]-code), using
- construction X with Varšamov bound [i] based on
(165−29, 165, large)-Net in Base 4 — Upper bound on s
There is no (136, 165, large)-net in base 4, because
- 27 times m-reduction [i] would yield (136, 138, large)-net in base 4, but