Best Known (66, 66+21, s)-Nets in Base 32
(66, 66+21, 104860)-Net over F32 — Constructive and digital
Digital (66, 87, 104860)-net over F32, using
- 321 times duplication [i] based on digital (65, 86, 104860)-net over F32, using
- net defined by OOA [i] based on linear OOA(3286, 104860, F32, 21, 21) (dual of [(104860, 21), 2201974, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3286, 1048601, F32, 21) (dual of [1048601, 1048515, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3286, 1048606, F32, 21) (dual of [1048606, 1048520, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(3281, 1048577, F32, 21) (dual of [1048577, 1048496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3257, 1048577, F32, 15) (dual of [1048577, 1048520, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(325, 29, F32, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3286, 1048606, F32, 21) (dual of [1048606, 1048520, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3286, 1048601, F32, 21) (dual of [1048601, 1048515, 22]-code), using
- net defined by OOA [i] based on linear OOA(3286, 104860, F32, 21, 21) (dual of [(104860, 21), 2201974, 22]-NRT-code), using
(66, 66+21, 209715)-Net in Base 32 — Constructive
(66, 87, 209715)-net in base 32, using
- 321 times duplication [i] based on (65, 86, 209715)-net in base 32, using
- net defined by OOA [i] based on OOA(3286, 209715, S32, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3286, 2097151, S32, 21), using
- discarding factors based on OA(3286, 2097155, S32, 21), using
- discarding parts of the base [i] based on linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- discarding factors based on OA(3286, 2097155, S32, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3286, 2097151, S32, 21), using
- net defined by OOA [i] based on OOA(3286, 209715, S32, 21, 21), using
(66, 66+21, 1048609)-Net over F32 — Digital
Digital (66, 87, 1048609)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3287, 1048609, F32, 21) (dual of [1048609, 1048522, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(3281, 1048576, F32, 21) (dual of [1048576, 1048495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(20) ⊂ Ce(13) [i] based on
(66, 66+21, large)-Net in Base 32 — Upper bound on s
There is no (66, 87, large)-net in base 32, because
- 19 times m-reduction [i] would yield (66, 68, large)-net in base 32, but