Best Known (134−23, 134, s)-Nets in Base 4
(134−23, 134, 1542)-Net over F4 — Constructive and digital
Digital (111, 134, 1542)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (31, 42, 514)-net over F4, using
- trace code for nets [i] based on digital (10, 21, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(10,256) in PG(20,16)) for nets [i] based on digital (0, 11, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(10,256) in PG(20,16)) for nets [i] based on digital (0, 11, 257)-net over F256, using
- trace code for nets [i] based on digital (10, 21, 257)-net over F16, using
- digital (69, 92, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 23, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 23, 257)-net over F256, using
- digital (31, 42, 514)-net over F4, using
(134−23, 134, 16440)-Net over F4 — Digital
Digital (111, 134, 16440)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4134, 16440, F4, 23) (dual of [16440, 16306, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(14) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(22) ⊂ Ce(14) [i] based on
(134−23, 134, large)-Net in Base 4 — Upper bound on s
There is no (111, 134, large)-net in base 4, because
- 21 times m-reduction [i] would yield (111, 113, large)-net in base 4, but