Best Known (70−35, 70, s)-Nets in Base 32
(70−35, 70, 218)-Net over F32 — Constructive and digital
Digital (35, 70, 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, 46, 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
(70−35, 70, 288)-Net in Base 32 — Constructive
(35, 70, 288)-net in base 32, using
- 21 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
(70−35, 70, 581)-Net over F32 — Digital
Digital (35, 70, 581)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3270, 581, F32, 35) (dual of [581, 511, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3270, 1030, F32, 35) (dual of [1030, 960, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(3269, 1025, F32, 35) (dual of [1025, 956, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(3265, 1025, F32, 33) (dual of [1025, 960, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 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 C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3270, 1030, F32, 35) (dual of [1030, 960, 36]-code), using
(70−35, 70, 297652)-Net in Base 32 — Upper bound on s
There is no (35, 70, 297653)-net in base 32, because
- 1 times m-reduction [i] would yield (35, 69, 297653)-net in base 32, but
- 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]