Best Known (39, 39+35, s)-Nets in Base 32
(39, 39+35, 240)-Net over F32 — Constructive and digital
Digital (39, 74, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 28, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 46, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 28, 120)-net over F32, using
(39, 39+35, 301)-Net in Base 32 — Constructive
(39, 74, 301)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- (21, 56, 257)-net in base 32, using
- base change [i] based on digital (0, 35, 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
- base change [i] based on digital (0, 35, 257)-net over F256, using
- digital (1, 18, 44)-net over F32, using
(39, 39+35, 892)-Net over F32 — Digital
Digital (39, 74, 892)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3274, 892, F32, 35) (dual of [892, 818, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3274, 1042, F32, 35) (dual of [1042, 968, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- linear OA(3269, 1025, F32, 35) (dual of [1025, 956, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(3257, 1025, F32, 29) (dual of [1025, 968, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3274, 1042, F32, 35) (dual of [1042, 968, 36]-code), using
(39, 39+35, 672773)-Net in Base 32 — Upper bound on s
There is no (39, 74, 672774)-net in base 32, because
- 1 times m-reduction [i] would yield (39, 73, 672774)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 75 153719 326126 766494 644142 908162 083930 119283 162096 471899 342800 262749 001524 072611 912721 265912 996560 678216 535104 > 3273 [i]