Best Known (54, 54+50, s)-Nets in Base 32
(54, 54+50, 240)-Net over F32 — Constructive and digital
Digital (54, 104, 240)-net over F32, using
- t-expansion [i] based on digital (51, 104, 240)-net over F32, using
- 5 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
- 5 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(54, 54+50, 513)-Net in Base 32 — Constructive
(54, 104, 513)-net in base 32, using
- t-expansion [i] based on (46, 104, 513)-net in base 32, using
- 4 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
- 4 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(54, 54+50, 1004)-Net over F32 — Digital
Digital (54, 104, 1004)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32104, 1004, F32, 50) (dual of [1004, 900, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(32104, 1050, F32, 50) (dual of [1050, 946, 51]-code), using
- construction X applied to Ce(49) ⊂ Ce(40) [i] based on
- linear OA(3296, 1024, F32, 50) (dual of [1024, 928, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [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(328, 26, F32, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,32)), using
- discarding factors / shortening the dual code based on linear OA(328, 32, F32, 8) (dual of [32, 24, 9]-code or 32-arc in PG(7,32)), using
- Reed–Solomon code RS(24,32) [i]
- discarding factors / shortening the dual code based on linear OA(328, 32, F32, 8) (dual of [32, 24, 9]-code or 32-arc in PG(7,32)), using
- construction X applied to Ce(49) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(32104, 1050, F32, 50) (dual of [1050, 946, 51]-code), using
(54, 54+50, 599347)-Net in Base 32 — Upper bound on s
There is no (54, 104, 599348)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 3 432457 488295 646628 038832 411400 183817 484190 783746 072856 030332 334800 399128 914973 690867 321237 851604 530605 601541 438550 928357 502553 774399 129464 141232 775996 993500 > 32104 [i]