Best Known (40−10, 40, s)-Nets in Base 32
(40−10, 40, 209719)-Net over F32 — Constructive and digital
Digital (30, 40, 209719)-net over F32, using
- net defined by OOA [i] based on linear OOA(3240, 209719, F32, 10, 10) (dual of [(209719, 10), 2097150, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3240, 1048595, F32, 10) (dual of [1048595, 1048555, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(323, 19, F32, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,32) or 19-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- OA 5-folding and stacking [i] based on linear OA(3240, 1048595, F32, 10) (dual of [1048595, 1048555, 11]-code), using
(40−10, 40, 419431)-Net in Base 32 — Constructive
(30, 40, 419431)-net in base 32, using
- net defined by OOA [i] based on OOA(3240, 419431, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3240, 2097155, S32, 10), using
- discarding parts of the base [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- OA 5-folding and stacking [i] based on OA(3240, 2097155, S32, 10), using
(40−10, 40, 1048595)-Net over F32 — Digital
Digital (30, 40, 1048595)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3240, 1048595, F32, 10) (dual of [1048595, 1048555, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(323, 19, F32, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,32) or 19-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
(40−10, 40, large)-Net in Base 32 — Upper bound on s
There is no (30, 40, large)-net in base 32, because
- 8 times m-reduction [i] would yield (30, 32, large)-net in base 32, but