Best Known (69−34, 69, s)-Nets in Base 32
(69−34, 69, 218)-Net over F32 — Constructive and digital
Digital (35, 69, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 24, 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 (11, 45, 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 (7, 24, 98)-net over F32, using
(69−34, 69, 288)-Net in Base 32 — Constructive
(35, 69, 288)-net in base 32, using
- 22 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 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, 65, 288)-net over F128, using
(69−34, 69, 637)-Net over F32 — Digital
Digital (35, 69, 637)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3269, 637, F32, 34) (dual of [637, 568, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3269, 1038, F32, 34) (dual of [1038, 969, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3264, 1024, F32, 34) (dual of [1024, 960, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3255, 1024, F32, 28) (dual of [1024, 969, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(325, 14, F32, 5) (dual of [14, 9, 6]-code or 14-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3269, 1038, F32, 34) (dual of [1038, 969, 35]-code), using
(69−34, 69, 297652)-Net in Base 32 — Upper bound on s
There is no (35, 69, 297653)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 71 674851 213140 123899 659000 498886 799038 395796 414971 351200 583726 614056 886442 049751 723357 066717 440468 814804 > 3269 [i]