Best Known (67−35, 67, s)-Nets in Base 32
(67−35, 67, 196)-Net over F32 — Constructive and digital
Digital (32, 67, 196)-net over F32, using
- 1 times m-reduction [i] based on digital (32, 68, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 43, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 25, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(67−35, 67, 288)-Net in Base 32 — Constructive
(32, 67, 288)-net in base 32, using
- t-expansion [i] based on (31, 67, 288)-net in base 32, using
- 10 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
- base change [i] based on digital (9, 55, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 55, 288)-net over F128, using
- 10 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
(67−35, 67, 455)-Net over F32 — Digital
Digital (32, 67, 455)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3267, 455, F32, 2, 35) (dual of [(455, 2), 843, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3267, 514, F32, 2, 35) (dual of [(514, 2), 961, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3267, 1028, F32, 35) (dual of [1028, 961, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- 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(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(321, 4, F32, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- OOA 2-folding [i] based on linear OA(3267, 1028, F32, 35) (dual of [1028, 961, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(3267, 514, F32, 2, 35) (dual of [(514, 2), 961, 36]-NRT-code), using
(67−35, 67, 161467)-Net in Base 32 — Upper bound on s
There is no (32, 67, 161468)-net in base 32, because
- 1 times m-reduction [i] would yield (32, 66, 161468)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2187 464710 558848 145134 443641 030402 086836 235095 599612 954594 283191 834474 520604 410696 800270 495957 149949 > 3266 [i]