Best Known (62−21, 62, s)-Nets in Base 32
(62−21, 62, 3277)-Net over F32 — Constructive and digital
Digital (41, 62, 3277)-net over F32, using
- 321 times duplication [i] based on digital (40, 61, 3277)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 3277, F32, 21, 21) (dual of [(3277, 21), 68756, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3261, 32771, F32, 21) (dual of [32771, 32710, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(3261, 32768, F32, 21) (dual of [32768, 32707, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(3261, 32771, F32, 21) (dual of [32771, 32710, 22]-code), using
- net defined by OOA [i] based on linear OOA(3261, 3277, F32, 21, 21) (dual of [(3277, 21), 68756, 22]-NRT-code), using
(62−21, 62, 17376)-Net over F32 — Digital
Digital (41, 62, 17376)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3262, 17376, F32, 21) (dual of [17376, 17314, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3262, 32776, F32, 21) (dual of [32776, 32714, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3261, 32769, F32, 21) (dual of [32769, 32708, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3262, 32776, F32, 21) (dual of [32776, 32714, 22]-code), using
(62−21, 62, large)-Net in Base 32 — Upper bound on s
There is no (41, 62, large)-net in base 32, because
- 19 times m-reduction [i] would yield (41, 43, large)-net in base 32, but