Best Known (31, 31+16, s)-Nets in Base 32
(31, 31+16, 4096)-Net over F32 — Constructive and digital
Digital (31, 47, 4096)-net over F32, using
- 321 times duplication [i] based on digital (30, 46, 4096)-net over F32, using
- net defined by OOA [i] based on linear OOA(3246, 4096, F32, 16, 16) (dual of [(4096, 16), 65490, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using
- net defined by OOA [i] based on linear OOA(3246, 4096, F32, 16, 16) (dual of [(4096, 16), 65490, 17]-NRT-code), using
(31, 31+16, 17196)-Net over F32 — Digital
Digital (31, 47, 17196)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3247, 17196, F32, 16) (dual of [17196, 17149, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3247, 32775, F32, 16) (dual of [32775, 32728, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3247, 32775, F32, 16) (dual of [32775, 32728, 17]-code), using
(31, 31+16, large)-Net in Base 32 — Upper bound on s
There is no (31, 47, large)-net in base 32, because
- 14 times m-reduction [i] would yield (31, 33, large)-net in base 32, but