Best Known (60, 60+17, s)-Nets in Base 4
(60, 60+17, 1037)-Net over F4 — Constructive and digital
Digital (60, 77, 1037)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- 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
- digital (1, 9, 9)-net over F4, using
(60, 60+17, 2393)-Net over F4 — Digital
Digital (60, 77, 2393)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(477, 2393, F4, 17) (dual of [2393, 2316, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 4113, F4, 17) (dual of [4113, 4036, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(473, 4097, F4, 17) (dual of [4097, 4024, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(461, 4097, F4, 13) (dual of [4097, 4036, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(44, 16, F4, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,4)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(477, 4113, F4, 17) (dual of [4113, 4036, 18]-code), using
(60, 60+17, 657861)-Net in Base 4 — Upper bound on s
There is no (60, 77, 657862)-net in base 4, because
- 1 times m-reduction [i] would yield (60, 76, 657862)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5709 005877 976688 014055 779024 737811 740664 017880 > 476 [i]