Best Known (50, 50+16, s)-Nets in Base 32
(50, 50+16, 131075)-Net over F32 — Constructive and digital
Digital (50, 66, 131075)-net over F32, using
- 321 times duplication [i] based on digital (49, 65, 131075)-net over F32, using
- net defined by OOA [i] based on linear OOA(3265, 131075, F32, 16, 16) (dual of [(131075, 16), 2097135, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3265, 1048600, F32, 16) (dual of [1048600, 1048535, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(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(15) ⊂ Ce(10) [i] based on
- OA 8-folding and stacking [i] based on linear OA(3265, 1048600, F32, 16) (dual of [1048600, 1048535, 17]-code), using
- net defined by OOA [i] based on linear OOA(3265, 131075, F32, 16, 16) (dual of [(131075, 16), 2097135, 17]-NRT-code), using
(50, 50+16, 262144)-Net in Base 32 — Constructive
(50, 66, 262144)-net in base 32, using
- 321 times duplication [i] based on (49, 65, 262144)-net in base 32, using
- net defined by OOA [i] based on OOA(3265, 262144, S32, 16, 16), using
- OA 8-folding and stacking [i] based on OA(3265, 2097152, S32, 16), using
- discarding factors based on OA(3265, 2097155, S32, 16), using
- discarding parts of the base [i] based on linear OA(12846, 2097155, F128, 16) (dual of [2097155, 2097109, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- discarding parts of the base [i] based on linear OA(12846, 2097155, F128, 16) (dual of [2097155, 2097109, 17]-code), using
- discarding factors based on OA(3265, 2097155, S32, 16), using
- OA 8-folding and stacking [i] based on OA(3265, 2097152, S32, 16), using
- net defined by OOA [i] based on OOA(3265, 262144, S32, 16, 16), using
(50, 50+16, 1048605)-Net over F32 — Digital
Digital (50, 66, 1048605)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3266, 1048605, F32, 16) (dual of [1048605, 1048539, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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(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(15) ⊂ Ce(9) [i] based on
(50, 50+16, large)-Net in Base 32 — Upper bound on s
There is no (50, 66, large)-net in base 32, because
- 14 times m-reduction [i] would yield (50, 52, large)-net in base 32, but