Best Known (54, 54+51, s)-Nets in Base 32
(54, 54+51, 240)-Net over F32 — Constructive and digital
Digital (54, 105, 240)-net over F32, using
- t-expansion [i] based on digital (51, 105, 240)-net over F32, using
- 4 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
- 4 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(54, 54+51, 513)-Net in Base 32 — Constructive
(54, 105, 513)-net in base 32, using
- t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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
- 3 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(54, 54+51, 942)-Net over F32 — Digital
Digital (54, 105, 942)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32105, 942, F32, 51) (dual of [942, 837, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(32105, 1047, F32, 51) (dual of [1047, 942, 52]-code), using
- construction X applied to Ce(50) ⊂ Ce(42) [i] based on
- linear OA(3298, 1024, F32, 51) (dual of [1024, 926, 52]-code), using an extension Ce(50) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,50], and designed minimum distance d ≥ |I|+1 = 51 [i]
- linear OA(3282, 1024, F32, 43) (dual of [1024, 942, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(50) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(32105, 1047, F32, 51) (dual of [1047, 942, 52]-code), using
(54, 54+51, 599347)-Net in Base 32 — Upper bound on s
There is no (54, 105, 599348)-net in base 32, because
- 1 times m-reduction [i] would yield (54, 104, 599348)-net in base 32, but
- 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]