Best Known (23, 23+19, s)-Nets in Base 32
(23, 23+19, 196)-Net over F32 — Constructive and digital
Digital (23, 42, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 26, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 16, 98)-net over F32, using
(23, 23+19, 322)-Net in Base 32 — Constructive
(23, 42, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (2, 11, 65)-net in base 32, using
- 1 times m-reduction [i] based on (2, 12, 65)-net in base 32, using
- base change [i] based on digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 10, 65)-net over F64, using
- 1 times m-reduction [i] based on (2, 12, 65)-net in base 32, using
- (12, 31, 257)-net in base 32, using
- 1 times m-reduction [i] based on (12, 32, 257)-net in base 32, using
- base change [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- 1 times m-reduction [i] based on (12, 32, 257)-net in base 32, using
- (2, 11, 65)-net in base 32, using
(23, 23+19, 980)-Net over F32 — Digital
Digital (23, 42, 980)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 980, F32, 19) (dual of [980, 938, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 1042, F32, 19) (dual of [1042, 1000, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(3237, 1025, F32, 19) (dual of [1025, 988, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(3225, 1025, F32, 13) (dual of [1025, 1000, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 1042, F32, 19) (dual of [1042, 1000, 20]-code), using
(23, 23+19, 962015)-Net in Base 32 — Upper bound on s
There is no (23, 42, 962016)-net in base 32, because
- 1 times m-reduction [i] would yield (23, 41, 962016)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 51 422158 137121 781267 687123 408498 946494 025409 045995 058818 006605 > 3241 [i]