Best Known (53, 53+25, s)-Nets in Base 32
(53, 53+25, 2732)-Net over F32 — Constructive and digital
Digital (53, 78, 2732)-net over F32, using
- 321 times duplication [i] based on digital (52, 77, 2732)-net over F32, using
- net defined by OOA [i] based on linear OOA(3277, 2732, F32, 25, 25) (dual of [(2732, 25), 68223, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3277, 32785, F32, 25) (dual of [32785, 32708, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3277, 32787, F32, 25) (dual of [32787, 32710, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3273, 32768, F32, 25) (dual of [32768, 32695, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3258, 32768, F32, 20) (dual of [32768, 32710, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3277, 32787, F32, 25) (dual of [32787, 32710, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3277, 32785, F32, 25) (dual of [32785, 32708, 26]-code), using
- net defined by OOA [i] based on linear OOA(3277, 2732, F32, 25, 25) (dual of [(2732, 25), 68223, 26]-NRT-code), using
(53, 53+25, 32792)-Net over F32 — Digital
Digital (53, 78, 32792)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3278, 32792, F32, 25) (dual of [32792, 32714, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3273, 32769, F32, 25) (dual of [32769, 32696, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(325, 23, F32, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
(53, 53+25, large)-Net in Base 32 — Upper bound on s
There is no (53, 78, large)-net in base 32, because
- 23 times m-reduction [i] would yield (53, 55, large)-net in base 32, but