Best Known (57, 57+18, s)-Nets in Base 32
(57, 57+18, 116512)-Net over F32 — Constructive and digital
Digital (57, 75, 116512)-net over F32, using
- net defined by OOA [i] based on linear OOA(3275, 116512, F32, 18, 18) (dual of [(116512, 18), 2097141, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3275, 1048608, F32, 18) (dual of [1048608, 1048533, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3275, 1048609, F32, 18) (dual of [1048609, 1048534, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3241, 1048576, F32, 11) (dual of [1048576, 1048535, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(17) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3275, 1048609, F32, 18) (dual of [1048609, 1048534, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3275, 1048608, F32, 18) (dual of [1048608, 1048533, 19]-code), using
(57, 57+18, 233017)-Net in Base 32 — Constructive
(57, 75, 233017)-net in base 32, using
- 322 times duplication [i] based on (55, 73, 233017)-net in base 32, using
- net defined by OOA [i] based on OOA(3273, 233017, S32, 18, 18), using
- OA 9-folding and stacking [i] based on OA(3273, 2097153, S32, 18), using
- discarding factors based on OA(3273, 2097155, S32, 18), using
- discarding parts of the base [i] based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding parts of the base [i] based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- discarding factors based on OA(3273, 2097155, S32, 18), using
- OA 9-folding and stacking [i] based on OA(3273, 2097153, S32, 18), using
- net defined by OOA [i] based on OOA(3273, 233017, S32, 18, 18), using
(57, 57+18, 1048609)-Net over F32 — Digital
Digital (57, 75, 1048609)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3275, 1048609, F32, 18) (dual of [1048609, 1048534, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3241, 1048576, F32, 11) (dual of [1048576, 1048535, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(17) ⊂ Ce(10) [i] based on
(57, 57+18, large)-Net in Base 32 — Upper bound on s
There is no (57, 75, large)-net in base 32, because
- 16 times m-reduction [i] would yield (57, 59, large)-net in base 32, but