Best Known (50, 50+45, s)-Nets in Base 32
(50, 50+45, 240)-Net over F32 — Constructive and digital
Digital (50, 95, 240)-net over F32, using
- 11 times m-reduction [i] based on digital (50, 106, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 39, 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, 67, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 39, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(50, 50+45, 513)-Net in Base 32 — Constructive
(50, 95, 513)-net in base 32, using
- t-expansion [i] based on (46, 95, 513)-net in base 32, using
- 13 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
- 13 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(50, 50+45, 1043)-Net over F32 — Digital
Digital (50, 95, 1043)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3295, 1043, F32, 45) (dual of [1043, 948, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3295, 1053, F32, 45) (dual of [1053, 958, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(34) [i] based on
- linear OA(3286, 1024, F32, 45) (dual of [1024, 938, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(3266, 1024, F32, 35) (dual of [1024, 958, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(329, 29, F32, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,32)), using
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- Reed–Solomon code RS(23,32) [i]
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- construction X applied to Ce(44) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3295, 1053, F32, 45) (dual of [1053, 958, 46]-code), using
(50, 50+45, 788098)-Net in Base 32 — Upper bound on s
There is no (50, 95, 788099)-net in base 32, because
- 1 times m-reduction [i] would yield (50, 94, 788099)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 3048 622511 268769 043974 972737 181382 842297 900464 123814 000035 891836 553412 333202 829146 344936 913495 117432 176786 957833 942127 492482 281905 475167 628064 > 3294 [i]