Best Known (25, 25+21, s)-Nets in Base 32
(25, 25+21, 196)-Net over F32 — Constructive and digital
Digital (25, 46, 196)-net over F32, using
- 1 times m-reduction [i] based on digital (25, 47, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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, 29, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 18, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(25, 25+21, 301)-Net in Base 32 — Constructive
(25, 46, 301)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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
- (14, 35, 257)-net in base 32, using
- 1 times m-reduction [i] based on (14, 36, 257)-net in base 32, using
- base change [i] based on (8, 30, 257)-net in base 64, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on (8, 30, 257)-net in base 64, using
- 1 times m-reduction [i] based on (14, 36, 257)-net in base 32, using
- digital (1, 11, 44)-net over F32, using
(25, 25+21, 930)-Net over F32 — Digital
Digital (25, 46, 930)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3246, 930, F32, 21) (dual of [930, 884, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 1042, F32, 21) (dual of [1042, 996, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(3241, 1025, F32, 21) (dual of [1025, 984, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3229, 1025, F32, 15) (dual of [1025, 996, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3246, 1042, F32, 21) (dual of [1042, 996, 22]-code), using
(25, 25+21, 866536)-Net in Base 32 — Upper bound on s
There is no (25, 46, 866537)-net in base 32, because
- 1 times m-reduction [i] would yield (25, 45, 866537)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 53 919913 635213 879013 137631 798362 725624 379334 127018 527348 846172 705808 > 3245 [i]