Best Known (53, 53+16, s)-Nets in Base 32
(53, 53+16, 131105)-Net over F32 — Constructive and digital
Digital (53, 69, 131105)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (45, 61, 131072)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 131072, F32, 16, 16) (dual of [(131072, 16), 2097091, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using
- net defined by OOA [i] based on linear OOA(3261, 131072, F32, 16, 16) (dual of [(131072, 16), 2097091, 17]-NRT-code), using
- digital (0, 8, 33)-net over F32, using
(53, 53+16, 262145)-Net in Base 32 — Constructive
(53, 69, 262145)-net in base 32, using
- 322 times duplication [i] based on (51, 67, 262145)-net in base 32, using
- net defined by OOA [i] based on OOA(3267, 262145, S32, 16, 16), using
- OA 8-folding and stacking [i] based on OA(3267, 2097160, S32, 16), using
- 1 times code embedding in larger space [i] based on OA(3266, 2097159, S32, 16), using
- discarding parts of the base [i] based on linear OA(12847, 2097159, F128, 16) (dual of [2097159, 2097112, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- 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(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(12847, 2097159, F128, 16) (dual of [2097159, 2097112, 17]-code), using
- 1 times code embedding in larger space [i] based on OA(3266, 2097159, S32, 16), using
- OA 8-folding and stacking [i] based on OA(3267, 2097160, S32, 16), using
- net defined by OOA [i] based on OOA(3267, 262145, S32, 16, 16), using
(53, 53+16, 1738188)-Net over F32 — Digital
Digital (53, 69, 1738188)-net over F32, using
(53, 53+16, large)-Net in Base 32 — Upper bound on s
There is no (53, 69, large)-net in base 32, because
- 14 times m-reduction [i] would yield (53, 55, large)-net in base 32, but