Best Known (55, 55+24, s)-Nets in Base 32
(55, 55+24, 2733)-Net over F32 — Constructive and digital
Digital (55, 79, 2733)-net over F32, using
- 1 times m-reduction [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
(55, 55+24, 5462)-Net in Base 32 — Constructive
(55, 79, 5462)-net in base 32, using
- net defined by OOA [i] based on OOA(3279, 5462, S32, 24, 24), using
- OA 12-folding and stacking [i] based on OA(3279, 65544, S32, 24), using
- discarding parts of the base [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- OA 12-folding and stacking [i] based on OA(3279, 65544, S32, 24), using
(55, 55+24, 44985)-Net over F32 — Digital
Digital (55, 79, 44985)-net over F32, using
(55, 55+24, large)-Net in Base 32 — Upper bound on s
There is no (55, 79, large)-net in base 32, because
- 22 times m-reduction [i] would yield (55, 57, large)-net in base 32, but