Best Known (40, 40+32, s)-Nets in Base 32
(40, 40+32, 240)-Net over F32 — Constructive and digital
Digital (40, 72, 240)-net over F32, using
- 4 times m-reduction [i] based on digital (40, 76, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 29, 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, 47, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 29, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(40, 40+32, 513)-Net in Base 32 — Constructive
(40, 72, 513)-net in base 32, using
- base change [i] based on digital (28, 60, 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
(40, 40+32, 1275)-Net over F32 — Digital
Digital (40, 72, 1275)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3272, 1275, F32, 32) (dual of [1275, 1203, 33]-code), using
- 242 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0, 1, 69 times 0, 1, 111 times 0) [i] based on linear OA(3262, 1023, F32, 32) (dual of [1023, 961, 33]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 242 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0, 1, 69 times 0, 1, 111 times 0) [i] based on linear OA(3262, 1023, F32, 32) (dual of [1023, 961, 33]-code), using
(40, 40+32, 1301215)-Net in Base 32 — Upper bound on s
There is no (40, 72, 1301216)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 348554 440763 228857 090715 869755 974853 866845 030367 535749 771683 261346 833880 128425 933140 190413 503747 066079 480671 > 3272 [i]