Best Known (57−19, 57, s)-Nets in Base 32
(57−19, 57, 3642)-Net over F32 — Constructive and digital
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
(57−19, 57, 21016)-Net over F32 — Digital
Digital (38, 57, 21016)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3257, 21016, F32, 19) (dual of [21016, 20959, 20]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3257, 32779, F32, 19) (dual of [32779, 32722, 20]-code), using
(57−19, 57, large)-Net in Base 32 — Upper bound on s
There is no (38, 57, large)-net in base 32, because
- 17 times m-reduction [i] would yield (38, 40, large)-net in base 32, but