Best Known (40, 59, s)-Nets in Base 32
(40, 59, 3642)-Net over F32 — Constructive and digital
Digital (40, 59, 3642)-net over F32, using
- 322 times duplication [i] based on digital (38, 57, 3642)-net over F32, using
- net defined by OOA [i] based on linear OOA(3257, 3642, F32, 19, 19) (dual of [(3642, 19), 69141, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3257, 32779, F32, 19) (dual of [32779, 32722, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3255, 32768, F32, 19) (dual of [32768, 32713, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(322, 11, F32, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(3257, 32779, F32, 19) (dual of [32779, 32722, 20]-code), using
- net defined by OOA [i] based on linear OOA(3257, 3642, F32, 19, 19) (dual of [(3642, 19), 69141, 20]-NRT-code), using
(40, 59, 31600)-Net over F32 — Digital
Digital (40, 59, 31600)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3259, 31600, F32, 19) (dual of [31600, 31541, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3259, 32787, F32, 19) (dual of [32787, 32728, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3255, 32768, F32, 19) (dual of [32768, 32713, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3259, 32787, F32, 19) (dual of [32787, 32728, 20]-code), using
(40, 59, large)-Net in Base 32 — Upper bound on s
There is no (40, 59, large)-net in base 32, because
- 17 times m-reduction [i] would yield (40, 42, large)-net in base 32, but