Best Known (108, 137, s)-Nets in Base 4
(108, 137, 1049)-Net over F4 — Constructive and digital
Digital (108, 137, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (87, 116, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 29, 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, 29, 257)-net over F256, using
- digital (7, 21, 21)-net over F4, using
(108, 137, 3904)-Net over F4 — Digital
Digital (108, 137, 3904)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4137, 3904, F4, 29) (dual of [3904, 3767, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4137, 4114, F4, 29) (dual of [4114, 3977, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(4133, 4097, F4, 29) (dual of [4097, 3964, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(4109, 4097, F4, 25) (dual of [4097, 3988, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4137, 4114, F4, 29) (dual of [4114, 3977, 30]-code), using
(108, 137, 1422053)-Net in Base 4 — Upper bound on s
There is no (108, 137, 1422054)-net in base 4, because
- 1 times m-reduction [i] would yield (108, 136, 1422054)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7588 609298 707068 713463 019190 752508 038909 740282 325136 113525 667230 777870 815472 127824 > 4136 [i]