Best Known (31, 31+14, s)-Nets in Base 32
(31, 31+14, 4684)-Net over F32 — Constructive and digital
Digital (31, 45, 4684)-net over F32, using
- net defined by OOA [i] based on linear OOA(3245, 4684, F32, 14, 14) (dual of [(4684, 14), 65531, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3245, 32788, F32, 14) (dual of [32788, 32743, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 32791, F32, 14) (dual of [32791, 32746, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3222, 32768, F32, 8) (dual of [32768, 32746, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(13) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 32791, F32, 14) (dual of [32791, 32746, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3245, 32788, F32, 14) (dual of [32788, 32743, 15]-code), using
(31, 31+14, 9363)-Net in Base 32 — Constructive
(31, 45, 9363)-net in base 32, using
- net defined by OOA [i] based on OOA(3245, 9363, S32, 14, 14), using
- OA 7-folding and stacking [i] based on OA(3245, 65541, S32, 14), using
- discarding parts of the base [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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 Ce(13) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- OA 7-folding and stacking [i] based on OA(3245, 65541, S32, 14), using
(31, 31+14, 32791)-Net over F32 — Digital
Digital (31, 45, 32791)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 32791, F32, 14) (dual of [32791, 32746, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3222, 32768, F32, 8) (dual of [32768, 32746, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(13) ⊂ Ce(7) [i] based on
(31, 31+14, large)-Net in Base 32 — Upper bound on s
There is no (31, 45, large)-net in base 32, because
- 12 times m-reduction [i] would yield (31, 33, large)-net in base 32, but