Best Known (82−19, 82, s)-Nets in Base 32
(82−19, 82, 116541)-Net over F32 — Constructive and digital
Digital (63, 82, 116541)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (54, 73, 116508)-net over F32, using
- net defined by OOA [i] based on linear OOA(3273, 116508, F32, 19, 19) (dual of [(116508, 19), 2213579, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3273, 1048573, F32, 19) (dual of [1048573, 1048500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3273, 1048573, F32, 19) (dual of [1048573, 1048500, 20]-code), using
- net defined by OOA [i] based on linear OOA(3273, 116508, F32, 19, 19) (dual of [(116508, 19), 2213579, 20]-NRT-code), using
- digital (0, 9, 33)-net over F32, using
(82−19, 82, 233018)-Net in Base 32 — Constructive
(63, 82, 233018)-net in base 32, using
- 322 times duplication [i] based on (61, 80, 233018)-net in base 32, using
- net defined by OOA [i] based on OOA(3280, 233018, S32, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3280, 2097163, S32, 19), using
- discarding parts of the base [i] based on linear OA(12857, 2097163, F128, 19) (dual of [2097163, 2097106, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12857, 2097163, F128, 19) (dual of [2097163, 2097106, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on OA(3280, 2097163, S32, 19), using
- net defined by OOA [i] based on OOA(3280, 233018, S32, 19, 19), using
(82−19, 82, 1752124)-Net over F32 — Digital
Digital (63, 82, 1752124)-net over F32, using
(82−19, 82, large)-Net in Base 32 — Upper bound on s
There is no (63, 82, large)-net in base 32, because
- 17 times m-reduction [i] would yield (63, 65, large)-net in base 32, but