Best Known (82−26, 82, s)-Nets in Base 32
(82−26, 82, 2522)-Net over F32 — Constructive and digital
Digital (56, 82, 2522)-net over F32, using
- 1 times m-reduction [i] based on digital (56, 83, 2522)-net over F32, using
- net defined by OOA [i] based on linear OOA(3283, 2522, F32, 27, 27) (dual of [(2522, 27), 68011, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3283, 32787, F32, 27) (dual of [32787, 32704, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3264, 32768, F32, 22) (dual of [32768, 32704, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(3283, 32787, F32, 27) (dual of [32787, 32704, 28]-code), using
- net defined by OOA [i] based on linear OOA(3283, 2522, F32, 27, 27) (dual of [(2522, 27), 68011, 28]-NRT-code), using
(82−26, 82, 5041)-Net in Base 32 — Constructive
(56, 82, 5041)-net in base 32, using
- net defined by OOA [i] based on OOA(3282, 5041, S32, 26, 26), using
- OA 13-folding and stacking [i] based on OA(3282, 65533, S32, 26), using
- discarding factors based on OA(3282, 65538, S32, 26), using
- discarding parts of the base [i] based on linear OA(25651, 65538, F256, 26) (dual of [65538, 65487, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- discarding parts of the base [i] based on linear OA(25651, 65538, F256, 26) (dual of [65538, 65487, 27]-code), using
- discarding factors based on OA(3282, 65538, S32, 26), using
- OA 13-folding and stacking [i] based on OA(3282, 65533, S32, 26), using
(82−26, 82, 32795)-Net over F32 — Digital
Digital (56, 82, 32795)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3282, 32795, F32, 26) (dual of [32795, 32713, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(3276, 32768, F32, 26) (dual of [32768, 32692, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(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(25) ⊂ Ce(18) [i] based on
(82−26, 82, large)-Net in Base 32 — Upper bound on s
There is no (56, 82, large)-net in base 32, because
- 24 times m-reduction [i] would yield (56, 58, large)-net in base 32, but