Best Known (72−22, 72, s)-Nets in Base 32
(72−22, 72, 2981)-Net over F32 — Constructive and digital
Digital (50, 72, 2981)-net over F32, using
- t-expansion [i] based on digital (49, 72, 2981)-net over F32, using
- net defined by OOA [i] based on linear OOA(3272, 2981, F32, 23, 23) (dual of [(2981, 23), 68491, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3272, 32792, F32, 23) (dual of [32792, 32720, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(3267, 32769, F32, 23) (dual of [32769, 32702, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,11]) ⊂ C([0,8]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(3272, 32792, F32, 23) (dual of [32792, 32720, 24]-code), using
- net defined by OOA [i] based on linear OOA(3272, 2981, F32, 23, 23) (dual of [(2981, 23), 68491, 24]-NRT-code), using
(72−22, 72, 5958)-Net in Base 32 — Constructive
(50, 72, 5958)-net in base 32, using
- base change [i] based on digital (23, 45, 5958)-net over F256, using
- 1 times m-reduction [i] based on digital (23, 46, 5958)-net over F256, using
- net defined by OOA [i] based on linear OOA(25646, 5958, F256, 23, 23) (dual of [(5958, 23), 136988, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(25646, 65539, F256, 23) (dual of [65539, 65493, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25646, 65542, F256, 23) (dual of [65542, 65496, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25646, 65542, F256, 23) (dual of [65542, 65496, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(25646, 65539, F256, 23) (dual of [65539, 65493, 24]-code), using
- net defined by OOA [i] based on linear OOA(25646, 5958, F256, 23, 23) (dual of [(5958, 23), 136988, 24]-NRT-code), using
- 1 times m-reduction [i] based on digital (23, 46, 5958)-net over F256, using
(72−22, 72, 40528)-Net over F32 — Digital
Digital (50, 72, 40528)-net over F32, using
(72−22, 72, large)-Net in Base 32 — Upper bound on s
There is no (50, 72, large)-net in base 32, because
- 20 times m-reduction [i] would yield (50, 52, large)-net in base 32, but