Best Known (53−18, 53, s)-Nets in Base 32
(53−18, 53, 3641)-Net over F32 — Constructive and digital
Digital (35, 53, 3641)-net over F32, using
- 321 times duplication [i] based on digital (34, 52, 3641)-net over F32, using
- net defined by OOA [i] based on linear OOA(3252, 3641, F32, 18, 18) (dual of [(3641, 18), 65486, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3252, 32769, F32, 18) (dual of [32769, 32717, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3252, 32771, F32, 18) (dual of [32771, 32719, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(3252, 32768, F32, 18) (dual of [32768, 32716, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3249, 32768, F32, 17) (dual of [32768, 32719, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3252, 32771, F32, 18) (dual of [32771, 32719, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3252, 32769, F32, 18) (dual of [32769, 32717, 19]-code), using
- net defined by OOA [i] based on linear OOA(3252, 3641, F32, 18, 18) (dual of [(3641, 18), 65486, 19]-NRT-code), using
(53−18, 53, 17089)-Net over F32 — Digital
Digital (35, 53, 17089)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3253, 17089, F32, 18) (dual of [17089, 17036, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3253, 32775, F32, 18) (dual of [32775, 32722, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(3252, 32768, F32, 18) (dual of [32768, 32716, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3253, 32775, F32, 18) (dual of [32775, 32722, 19]-code), using
(53−18, 53, large)-Net in Base 32 — Upper bound on s
There is no (35, 53, large)-net in base 32, because
- 16 times m-reduction [i] would yield (35, 37, large)-net in base 32, but