Best Known (88−42, 88, s)-Nets in Base 32
(88−42, 88, 240)-Net over F32 — Constructive and digital
Digital (46, 88, 240)-net over F32, using
- 6 times m-reduction [i] based on digital (46, 94, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 35, 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, 59, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 35, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(88−42, 88, 513)-Net in Base 32 — Constructive
(46, 88, 513)-net in base 32, using
- 20 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
(88−42, 88, 937)-Net over F32 — Digital
Digital (46, 88, 937)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3288, 937, F32, 42) (dual of [937, 849, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3288, 1049, F32, 42) (dual of [1049, 961, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(32) [i] based on
- linear OA(3280, 1024, F32, 42) (dual of [1024, 944, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- 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]
- linear OA(328, 25, F32, 8) (dual of [25, 17, 9]-code or 25-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(41) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(3288, 1049, F32, 42) (dual of [1049, 961, 43]-code), using
(88−42, 88, 568090)-Net in Base 32 — Upper bound on s
There is no (46, 88, 568091)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 839220 760889 431087 371836 947399 003780 646354 278821 915392 330350 742908 083495 655313 886820 192256 302887 119123 209102 618275 810923 192816 204684 > 3288 [i]