Best Known (65, 65+27, s)-Nets in Base 32
(65, 65+27, 2553)-Net over F32 — Constructive and digital
Digital (65, 92, 2553)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (52, 79, 2520)-net over F32, using
- net defined by OOA [i] based on linear OOA(3279, 2520, F32, 27, 27) (dual of [(2520, 27), 67961, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3279, 32761, F32, 27) (dual of [32761, 32682, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3279, 32761, F32, 27) (dual of [32761, 32682, 28]-code), using
- net defined by OOA [i] based on linear OOA(3279, 2520, F32, 27, 27) (dual of [(2520, 27), 67961, 28]-NRT-code), using
- digital (0, 13, 33)-net over F32, using
(65, 65+27, 5042)-Net in Base 32 — Constructive
(65, 92, 5042)-net in base 32, using
- 321 times duplication [i] based on (64, 91, 5042)-net in base 32, using
- base change [i] based on (38, 65, 5042)-net in base 128, using
- 1281 times duplication [i] based on (37, 64, 5042)-net in base 128, using
- base change [i] based on digital (29, 56, 5042)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- base change [i] based on digital (29, 56, 5042)-net over F256, using
- 1281 times duplication [i] based on (37, 64, 5042)-net in base 128, using
- base change [i] based on (38, 65, 5042)-net in base 128, using
(65, 65+27, 72098)-Net over F32 — Digital
Digital (65, 92, 72098)-net over F32, using
(65, 65+27, large)-Net in Base 32 — Upper bound on s
There is no (65, 92, large)-net in base 32, because
- 25 times m-reduction [i] would yield (65, 67, large)-net in base 32, but