Best Known (68−36, 68, s)-Nets in Base 32
(68−36, 68, 196)-Net over F32 — Constructive and digital
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
(68−36, 68, 288)-Net in Base 32 — Constructive
(32, 68, 288)-net in base 32, using
- t-expansion [i] based on (31, 68, 288)-net in base 32, using
- 9 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
- 9 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
(68−36, 68, 419)-Net over F32 — Digital
Digital (32, 68, 419)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3268, 419, F32, 2, 36) (dual of [(419, 2), 770, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3268, 513, F32, 2, 36) (dual of [(513, 2), 958, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3268, 1026, F32, 36) (dual of [1026, 958, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- linear OA(3268, 1024, F32, 36) (dual of [1024, 956, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(3268, 1026, F32, 36) (dual of [1026, 958, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(3268, 513, F32, 2, 36) (dual of [(513, 2), 958, 37]-NRT-code), using
(68−36, 68, 118265)-Net in Base 32 — Upper bound on s
There is no (32, 68, 118266)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 239999 954647 139948 596602 330213 612380 310652 129605 644058 409917 669757 579724 882255 517758 340147 316118 625284 > 3268 [i]