Best Known (133, 133+28, s)-Nets in Base 4
(133, 133+28, 1272)-Net over F4 — Constructive and digital
Digital (133, 161, 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 (88, 116, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 29, 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, 29, 258)-net over F256, using
- digital (31, 45, 240)-net over F4, using
(133, 133+28, 16439)-Net over F4 — Digital
Digital (133, 161, 16439)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4161, 16439, F4, 28) (dual of [16439, 16278, 29]-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(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
(133, 133+28, large)-Net in Base 4 — Upper bound on s
There is no (133, 161, large)-net in base 4, because
- 26 times m-reduction [i] would yield (133, 135, large)-net in base 4, but