Best Known (56, 56+18, s)-Nets in Base 32
(56, 56+18, 116511)-Net over F32 — Constructive and digital
Digital (56, 74, 116511)-net over F32, using
- 321 times duplication [i] based on digital (55, 73, 116511)-net over F32, using
- net defined by OOA [i] based on linear OOA(3273, 116511, F32, 18, 18) (dual of [(116511, 18), 2097125, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3273, 1048599, F32, 18) (dual of [1048599, 1048526, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 1048600, F32, 18) (dual of [1048600, 1048527, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [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(3249, 1048576, F32, 13) (dual of [1048576, 1048527, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(324, 24, F32, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3273, 1048600, F32, 18) (dual of [1048600, 1048527, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3273, 1048599, F32, 18) (dual of [1048599, 1048526, 19]-code), using
- net defined by OOA [i] based on linear OOA(3273, 116511, F32, 18, 18) (dual of [(116511, 18), 2097125, 19]-NRT-code), using
(56, 56+18, 233017)-Net in Base 32 — Constructive
(56, 74, 233017)-net in base 32, using
- 321 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
(56, 56+18, 1048605)-Net over F32 — Digital
Digital (56, 74, 1048605)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3274, 1048605, F32, 18) (dual of [1048605, 1048531, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [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(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(17) ⊂ Ce(11) [i] based on
(56, 56+18, large)-Net in Base 32 — Upper bound on s
There is no (56, 74, large)-net in base 32, because
- 16 times m-reduction [i] would yield (56, 58, large)-net in base 32, but