Best Known (87, 110, s)-Nets in Base 4
(87, 110, 1049)-Net over F4 — Constructive and digital
Digital (87, 110, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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 (69, 92, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- digital (7, 18, 21)-net over F4, using
(87, 110, 3842)-Net over F4 — Digital
Digital (87, 110, 3842)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4110, 3842, F4, 23) (dual of [3842, 3732, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 4110, F4, 23) (dual of [4110, 4000, 24]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 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(497, 4097, F4, 21) (dual of [4097, 4000, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4110, 4110, F4, 23) (dual of [4110, 4000, 24]-code), using
(87, 110, 1512716)-Net in Base 4 — Upper bound on s
There is no (87, 110, 1512717)-net in base 4, because
- 1 times m-reduction [i] would yield (87, 109, 1512717)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 421250 099625 157688 802863 153225 732080 303942 036819 474819 725234 282460 > 4109 [i]