Best Known (109, 109+27, s)-Nets in Base 4
(109, 109+27, 1076)-Net over F4 — Constructive and digital
Digital (109, 136, 1076)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (15, 28, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 14, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 14, 24)-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 (15, 28, 48)-net over F4, using
(109, 109+27, 4998)-Net over F4 — Digital
Digital (109, 136, 4998)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4136, 4998, F4, 27) (dual of [4998, 4862, 28]-code), using
- 881 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, 1, 188 times 0, 1, 225 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
- 881 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, 1, 188 times 0, 1, 225 times 0) [i] based on linear OA(4121, 4102, F4, 27) (dual of [4102, 3981, 28]-code), using
(109, 109+27, 3376331)-Net in Base 4 — Upper bound on s
There is no (109, 136, 3376332)-net in base 4, because
- 1 times m-reduction [i] would yield (109, 135, 3376332)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1897 143393 929919 018928 664850 826864 882675 604228 761786 647932 807666 977446 717263 419120 > 4135 [i]