Best Known (42, 42+33, s)-Nets in Base 32
(42, 42+33, 240)-Net over F32 — Constructive and digital
Digital (42, 75, 240)-net over F32, using
- 7 times m-reduction [i] based on digital (42, 82, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 31, 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, 51, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 31, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(42, 42+33, 513)-Net in Base 32 — Constructive
(42, 75, 513)-net in base 32, using
- 9 times m-reduction [i] based on (42, 84, 513)-net in base 32, using
- base change [i] based on digital (28, 70, 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, 70, 513)-net over F64, using
(42, 42+33, 1408)-Net over F32 — Digital
Digital (42, 75, 1408)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3275, 1408, F32, 33) (dual of [1408, 1333, 34]-code), using
- 372 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 35 times 0, 1, 65 times 0, 1, 100 times 0, 1, 133 times 0) [i] based on linear OA(3263, 1024, F32, 33) (dual of [1024, 961, 34]-code), using
- an extension Ce(32) of 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]
- 372 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 35 times 0, 1, 65 times 0, 1, 100 times 0, 1, 133 times 0) [i] based on linear OA(3263, 1024, F32, 33) (dual of [1024, 961, 34]-code), using
(42, 42+33, 2006753)-Net in Base 32 — Upper bound on s
There is no (42, 75, 2006754)-net in base 32, because
- 1 times m-reduction [i] would yield (42, 74, 2006754)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2404 924582 857688 830959 994366 761286 019745 665618 932556 443928 297823 943151 206794 813368 290876 983541 552345 125148 693370 > 3274 [i]