Best Known (52, 52+49, s)-Nets in Base 32
(52, 52+49, 240)-Net over F32 — Constructive and digital
Digital (52, 101, 240)-net over F32, using
- t-expansion [i] based on digital (51, 101, 240)-net over F32, using
- 8 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 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, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
- 8 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(52, 52+49, 513)-Net in Base 32 — Constructive
(52, 101, 513)-net in base 32, using
- t-expansion [i] based on (46, 101, 513)-net in base 32, using
- 7 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 7 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(52, 52+49, 923)-Net over F32 — Digital
Digital (52, 101, 923)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32101, 923, F32, 49) (dual of [923, 822, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(32101, 1047, F32, 49) (dual of [1047, 946, 50]-code), using
- construction X applied to Ce(48) ⊂ Ce(40) [i] based on
- linear OA(3294, 1024, F32, 49) (dual of [1024, 930, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3278, 1024, F32, 41) (dual of [1024, 946, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(327, 23, F32, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- construction X applied to Ce(48) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(32101, 1047, F32, 49) (dual of [1047, 946, 50]-code), using
(52, 52+49, 590813)-Net in Base 32 — Upper bound on s
There is no (52, 101, 590814)-net in base 32, because
- 1 times m-reduction [i] would yield (52, 100, 590814)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 3 273483 731992 338137 349328 924184 942372 902809 971619 801483 909568 063501 710476 526525 866536 245179 500907 382647 396738 843911 060939 051663 025555 250123 140005 133863 > 32100 [i]