Best Known (109−26, 109, s)-Nets in Base 32
(109−26, 109, 80662)-Net over F32 — Constructive and digital
Digital (83, 109, 80662)-net over F32, using
- 322 times duplication [i] based on digital (81, 107, 80662)-net over F32, using
- net defined by OOA [i] based on linear OOA(32107, 80662, F32, 26, 26) (dual of [(80662, 26), 2097105, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(32107, 1048606, F32, 26) (dual of [1048606, 1048499, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(32107, 1048609, F32, 26) (dual of [1048609, 1048502, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(32101, 1048576, F32, 26) (dual of [1048576, 1048475, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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]
- linear OA(326, 33, F32, 6) (dual of [33, 27, 7]-code or 33-arc in PG(5,32)), using
- extended Reed–Solomon code RSe(27,32) [i]
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(32107, 1048609, F32, 26) (dual of [1048609, 1048502, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(32107, 1048606, F32, 26) (dual of [1048606, 1048499, 27]-code), using
- net defined by OOA [i] based on linear OOA(32107, 80662, F32, 26, 26) (dual of [(80662, 26), 2097105, 27]-NRT-code), using
(109−26, 109, 161320)-Net in Base 32 — Constructive
(83, 109, 161320)-net in base 32, using
- net defined by OOA [i] based on OOA(32109, 161320, S32, 26, 26), using
- OA 13-folding and stacking [i] based on OA(32109, 2097160, S32, 26), using
- 1 times code embedding in larger space [i] based on OA(32108, 2097159, S32, 26), using
- discarding parts of the base [i] based on linear OA(12877, 2097159, F128, 26) (dual of [2097159, 2097082, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(12876, 2097152, F128, 26) (dual of [2097152, 2097076, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(25) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(12877, 2097159, F128, 26) (dual of [2097159, 2097082, 27]-code), using
- 1 times code embedding in larger space [i] based on OA(32108, 2097159, S32, 26), using
- OA 13-folding and stacking [i] based on OA(32109, 2097160, S32, 26), using
(109−26, 109, 1198733)-Net over F32 — Digital
Digital (83, 109, 1198733)-net over F32, using
(109−26, 109, large)-Net in Base 32 — Upper bound on s
There is no (83, 109, large)-net in base 32, because
- 24 times m-reduction [i] would yield (83, 85, large)-net in base 32, but