Best Known (42−13, 42, s)-Nets in Base 32
(42−13, 42, 5465)-Net over F32 — Constructive and digital
Digital (29, 42, 5465)-net over F32, using
- net defined by OOA [i] based on linear OOA(3242, 5465, F32, 13, 13) (dual of [(5465, 13), 71003, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3242, 32791, F32, 13) (dual of [32791, 32749, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 32792, F32, 13) (dual of [32792, 32750, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- 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(3219, 32769, F32, 7) (dual of [32769, 32750, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,6]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 32792, F32, 13) (dual of [32792, 32750, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3242, 32791, F32, 13) (dual of [32791, 32749, 14]-code), using
(42−13, 42, 10923)-Net in Base 32 — Constructive
(29, 42, 10923)-net in base 32, using
- 321 times duplication [i] based on (28, 41, 10923)-net in base 32, using
- net defined by OOA [i] based on OOA(3241, 10923, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3241, 65539, S32, 13), using
- 1 times code embedding in larger space [i] based on OA(3240, 65538, S32, 13), using
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- 1 times code embedding in larger space [i] based on OA(3240, 65538, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3241, 65539, S32, 13), using
- net defined by OOA [i] based on OOA(3241, 10923, S32, 13, 13), using
(42−13, 42, 32792)-Net over F32 — Digital
Digital (29, 42, 32792)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 32792, F32, 13) (dual of [32792, 32750, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- 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(3219, 32769, F32, 7) (dual of [32769, 32750, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,6]) ⊂ C([0,3]) [i] based on
(42−13, 42, large)-Net in Base 32 — Upper bound on s
There is no (29, 42, large)-net in base 32, because
- 11 times m-reduction [i] would yield (29, 31, large)-net in base 32, but