Best Known (42, 42+19, s)-Nets in Base 32
(42, 42+19, 3643)-Net over F32 — Constructive and digital
Digital (42, 61, 3643)-net over F32, using
- 321 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(3260, 3643, F32, 19, 19) (dual of [(3643, 19), 69157, 20]-NRT-code), using
(42, 42+19, 7282)-Net in Base 32 — Constructive
(42, 61, 7282)-net in base 32, using
- net defined by OOA [i] based on OOA(3261, 7282, S32, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3261, 65539, S32, 19), using
- discarding factors based on OA(3261, 65542, S32, 19), using
- discarding parts of the base [i] based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- discarding parts of the base [i] based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- discarding factors based on OA(3261, 65542, S32, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3261, 65539, S32, 19), using
(42, 42+19, 32795)-Net over F32 — Digital
Digital (42, 61, 32795)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3261, 32795, F32, 19) (dual of [32795, 32734, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- linear OA(3255, 32768, F32, 19) (dual of [32768, 32713, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3234, 32768, F32, 12) (dual of [32768, 32734, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(326, 27, F32, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,32)), using
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- Reed–Solomon code RS(26,32) [i]
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
(42, 42+19, large)-Net in Base 32 — Upper bound on s
There is no (42, 61, large)-net in base 32, because
- 17 times m-reduction [i] would yield (42, 44, large)-net in base 32, but