Best Known (45, 59, s)-Nets in Base 4
(45, 59, 1028)-Net over F4 — Constructive and digital
Digital (45, 59, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (45, 60, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
(45, 59, 1128)-Net over F4 — Digital
Digital (45, 59, 1128)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(459, 1128, F4, 14) (dual of [1128, 1069, 15]-code), using
- 91 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 23 times 0, 1, 42 times 0) [i] based on linear OA(451, 1029, F4, 14) (dual of [1029, 978, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(451, 1024, F4, 14) (dual of [1024, 973, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(446, 1024, F4, 13) (dual of [1024, 978, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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(13) ⊂ Ce(12) [i] based on
- 91 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 23 times 0, 1, 42 times 0) [i] based on linear OA(451, 1029, F4, 14) (dual of [1029, 978, 15]-code), using
(45, 59, 133747)-Net in Base 4 — Upper bound on s
There is no (45, 59, 133748)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 332315 457950 461203 299445 283177 932148 > 459 [i]