Best Known (134−27, 134, s)-Nets in Base 4
(134−27, 134, 1062)-Net over F4 — Constructive and digital
Digital (107, 134, 1062)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 34)-net over F4, using
- trace code for nets [i] based on digital (0, 13, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- trace code for nets [i] based on digital (0, 13, 17)-net over F16, using
- digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 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
- trace code for nets [i] based on digital (0, 27, 257)-net over F256, using
- digital (13, 26, 34)-net over F4, using
(134−27, 134, 4581)-Net over F4 — Digital
Digital (107, 134, 4581)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4134, 4581, F4, 27) (dual of [4581, 4447, 28]-code), using
- 466 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0, 1, 74 times 0, 1, 108 times 0, 1, 147 times 0) [i] based on linear OA(4121, 4102, F4, 27) (dual of [4102, 3981, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4121, 4096, F4, 27) (dual of [4096, 3975, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4115, 4096, F4, 26) (dual of [4096, 3981, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 466 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0, 1, 74 times 0, 1, 108 times 0, 1, 147 times 0) [i] based on linear OA(4121, 4102, F4, 27) (dual of [4102, 3981, 28]-code), using
(134−27, 134, 2727847)-Net in Base 4 — Upper bound on s
There is no (107, 134, 2727848)-net in base 4, because
- 1 times m-reduction [i] would yield (107, 133, 2727848)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 118 571292 478570 820421 396843 931927 637017 681911 258064 149661 211958 532060 515964 067500 > 4133 [i]