Best Known (41, 60, s)-Nets in Base 32
(41, 60, 3643)-Net over F32 — Constructive and digital
Digital (41, 60, 3643)-net over F32, using
- net defined by OOA [i] based on linear OOA(3260, 3643, F32, 19, 19) (dual of [(3643, 19), 69157, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3260, 32788, F32, 19) (dual of [32788, 32728, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3260, 32792, F32, 19) (dual of [32792, 32732, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- 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(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3260, 32792, F32, 19) (dual of [32792, 32732, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3260, 32788, F32, 19) (dual of [32788, 32728, 20]-code), using
(41, 60, 7281)-Net in Base 32 — Constructive
(41, 60, 7281)-net in base 32, using
- net defined by OOA [i] based on OOA(3260, 7281, S32, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3260, 65530, S32, 19), using
- discarding factors based on OA(3260, 65538, S32, 19), using
- discarding parts of the base [i] based on linear OA(25637, 65538, F256, 19) (dual of [65538, 65501, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(25637, 65536, F256, 19) (dual of [65536, 65499, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(25637, 65538, F256, 19) (dual of [65538, 65501, 20]-code), using
- discarding factors based on OA(3260, 65538, S32, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3260, 65530, S32, 19), using
(41, 60, 32792)-Net over F32 — Digital
Digital (41, 60, 32792)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3260, 32792, F32, 19) (dual of [32792, 32732, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- 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(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,9]) ⊂ C([0,6]) [i] based on
(41, 60, large)-Net in Base 32 — Upper bound on s
There is no (41, 60, large)-net in base 32, because
- 17 times m-reduction [i] would yield (41, 43, large)-net in base 32, but