Best Known (45−15, 45, s)-Nets in Base 32
(45−15, 45, 4682)-Net over F32 — Constructive and digital
Digital (30, 45, 4682)-net over F32, using
- 321 times duplication [i] based on digital (29, 44, 4682)-net over F32, using
- net defined by OOA [i] based on linear OOA(3244, 4682, F32, 15, 15) (dual of [(4682, 15), 70186, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3244, 32775, F32, 15) (dual of [32775, 32731, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 32776, F32, 15) (dual of [32776, 32732, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 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(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 32776, F32, 15) (dual of [32776, 32732, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3244, 32775, F32, 15) (dual of [32775, 32731, 16]-code), using
- net defined by OOA [i] based on linear OOA(3244, 4682, F32, 15, 15) (dual of [(4682, 15), 70186, 16]-NRT-code), using
(45−15, 45, 22714)-Net over F32 — Digital
Digital (30, 45, 22714)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 22714, F32, 15) (dual of [22714, 22669, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 32779, F32, 15) (dual of [32779, 32734, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(3243, 32768, F32, 15) (dual of [32768, 32725, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3234, 32768, F32, 12) (dual of [32768, 32734, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 32779, F32, 15) (dual of [32779, 32734, 16]-code), using
(45−15, 45, large)-Net in Base 32 — Upper bound on s
There is no (30, 45, large)-net in base 32, because
- 13 times m-reduction [i] would yield (30, 32, large)-net in base 32, but