Best Known (58−14, 58, s)-Nets in Base 32
(58−14, 58, 149800)-Net over F32 — Constructive and digital
Digital (44, 58, 149800)-net over F32, using
- 321 times duplication [i] based on digital (43, 57, 149800)-net over F32, using
- net defined by OOA [i] based on linear OOA(3257, 149800, F32, 14, 14) (dual of [(149800, 14), 2097143, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3257, 1048600, F32, 14) (dual of [1048600, 1048543, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- 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(3233, 1048576, F32, 9) (dual of [1048576, 1048543, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(13) ⊂ Ce(8) [i] based on
- OA 7-folding and stacking [i] based on linear OA(3257, 1048600, F32, 14) (dual of [1048600, 1048543, 15]-code), using
- net defined by OOA [i] based on linear OOA(3257, 149800, F32, 14, 14) (dual of [(149800, 14), 2097143, 15]-NRT-code), using
(58−14, 58, 299594)-Net in Base 32 — Constructive
(44, 58, 299594)-net in base 32, using
- net defined by OOA [i] based on OOA(3258, 299594, S32, 14, 14), using
- OA 7-folding and stacking [i] based on OA(3258, 2097158, S32, 14), using
- discarding factors based on OA(3258, 2097159, S32, 14), using
- discarding parts of the base [i] based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- discarding factors based on OA(3258, 2097159, S32, 14), using
- OA 7-folding and stacking [i] based on OA(3258, 2097158, S32, 14), using
(58−14, 58, 1048605)-Net over F32 — Digital
Digital (44, 58, 1048605)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3258, 1048605, F32, 14) (dual of [1048605, 1048547, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- 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(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(13) ⊂ Ce(7) [i] based on
(58−14, 58, large)-Net in Base 32 — Upper bound on s
There is no (44, 58, large)-net in base 32, because
- 12 times m-reduction [i] would yield (44, 46, large)-net in base 32, but