Best Known (58−19, 58, s)-Nets in Base 32
(58−19, 58, 3642)-Net over F32 — Constructive and digital
Digital (39, 58, 3642)-net over F32, using
- 321 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
(58−19, 58, 25771)-Net over F32 — Digital
Digital (39, 58, 25771)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3258, 25771, F32, 19) (dual of [25771, 25713, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3258, 32784, F32, 19) (dual of [32784, 32726, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(3255, 32769, F32, 19) (dual of [32769, 32714, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(3243, 32769, F32, 15) (dual of [32769, 32726, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3258, 32784, F32, 19) (dual of [32784, 32726, 20]-code), using
(58−19, 58, large)-Net in Base 32 — Upper bound on s
There is no (39, 58, large)-net in base 32, because
- 17 times m-reduction [i] would yield (39, 41, large)-net in base 32, but