Best Known (51−26, 51, s)-Nets in Base 32
(51−26, 51, 174)-Net over F32 — Constructive and digital
Digital (25, 51, 174)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- digital (7, 33, 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 (5, 18, 76)-net over F32, using
(51−26, 51, 288)-Net in Base 32 — Constructive
(25, 51, 288)-net in base 32, using
- 5 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
- base change [i] based on digital (9, 40, 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, 40, 288)-net over F128, using
(51−26, 51, 478)-Net over F32 — Digital
Digital (25, 51, 478)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3251, 478, F32, 2, 26) (dual of [(478, 2), 905, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3251, 513, F32, 2, 26) (dual of [(513, 2), 975, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3251, 1026, F32, 26) (dual of [1026, 975, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3251, 1024, F32, 26) (dual of [1024, 973, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3249, 1024, F32, 25) (dual of [1024, 975, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(3251, 1026, F32, 26) (dual of [1026, 975, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(3251, 513, F32, 2, 26) (dual of [(513, 2), 975, 27]-NRT-code), using
(51−26, 51, 146840)-Net in Base 32 — Upper bound on s
There is no (25, 51, 146841)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 57900 704124 435695 514956 836918 997658 651383 936261 698758 433669 435482 780402 828912 > 3251 [i]