Best Known (53, 70, s)-Nets in Base 4
(53, 70, 1028)-Net over F4 — Constructive and digital
Digital (53, 70, 1028)-net over F4, using
- 42 times duplication [i] based on digital (51, 68, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 257)-net over F256, using
(53, 70, 1112)-Net over F4 — Digital
Digital (53, 70, 1112)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(470, 1112, F4, 17) (dual of [1112, 1042, 18]-code), using
- 78 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 19 times 0, 1, 29 times 0) [i] based on linear OA(461, 1025, F4, 17) (dual of [1025, 964, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 78 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 19 times 0, 1, 29 times 0) [i] based on linear OA(461, 1025, F4, 17) (dual of [1025, 964, 18]-code), using
(53, 70, 195579)-Net in Base 4 — Upper bound on s
There is no (53, 70, 195580)-net in base 4, because
- 1 times m-reduction [i] would yield (53, 69, 195580)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 348461 332108 122018 141044 584332 964896 997519 > 469 [i]