Best Known (56, 56+25, s)-Nets in Base 32
(56, 56+25, 2733)-Net over F32 — Constructive and digital
Digital (56, 81, 2733)-net over F32, using
- 321 times duplication [i] based on digital (55, 80, 2733)-net over F32, using
- net defined by OOA [i] based on linear OOA(3280, 2733, F32, 25, 25) (dual of [(2733, 25), 68245, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3280, 32797, F32, 25) (dual of [32797, 32717, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3280, 32800, F32, 25) (dual of [32800, 32720, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [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(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(327, 31, F32, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3280, 32800, F32, 25) (dual of [32800, 32720, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3280, 32797, F32, 25) (dual of [32797, 32717, 26]-code), using
- net defined by OOA [i] based on linear OOA(3280, 2733, F32, 25, 25) (dual of [(2733, 25), 68245, 26]-NRT-code), using
(56, 56+25, 5461)-Net in Base 32 — Constructive
(56, 81, 5461)-net in base 32, using
- 321 times duplication [i] based on (55, 80, 5461)-net in base 32, using
- base change [i] based on digital (25, 50, 5461)-net over F256, using
- 2561 times duplication [i] based on digital (24, 49, 5461)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
- 2561 times duplication [i] based on digital (24, 49, 5461)-net over F256, using
- base change [i] based on digital (25, 50, 5461)-net over F256, using
(56, 56+25, 38021)-Net over F32 — Digital
Digital (56, 81, 38021)-net over F32, using
(56, 56+25, large)-Net in Base 32 — Upper bound on s
There is no (56, 81, large)-net in base 32, because
- 23 times m-reduction [i] would yield (56, 58, large)-net in base 32, but