Best Known (27−7, 27, s)-Nets in Base 32
(27−7, 27, 349529)-Net over F32 — Constructive and digital
Digital (20, 27, 349529)-net over F32, using
- net defined by OOA [i] based on linear OOA(3227, 349529, F32, 7, 7) (dual of [(349529, 7), 2446676, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3227, 1048588, F32, 7) (dual of [1048588, 1048561, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 1048590, F32, 7) (dual of [1048590, 1048563, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(3225, 1048576, F32, 7) (dual of [1048576, 1048551, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 1048590, F32, 7) (dual of [1048590, 1048563, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3227, 1048588, F32, 7) (dual of [1048588, 1048561, 8]-code), using
(27−7, 27, 699051)-Net in Base 32 — Constructive
(20, 27, 699051)-net in base 32, using
- net defined by OOA [i] based on OOA(3227, 699051, S32, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(3227, 2097154, S32, 7), using
- discarding factors based on OA(3227, 2097155, S32, 7), using
- discarding parts of the base [i] based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- discarding factors based on OA(3227, 2097155, S32, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(3227, 2097154, S32, 7), using
(27−7, 27, 1048590)-Net over F32 — Digital
Digital (20, 27, 1048590)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 1048590, F32, 7) (dual of [1048590, 1048563, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(3225, 1048576, F32, 7) (dual of [1048576, 1048551, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
(27−7, 27, large)-Net in Base 32 — Upper bound on s
There is no (20, 27, large)-net in base 32, because
- 5 times m-reduction [i] would yield (20, 22, large)-net in base 32, but