Best Known (73−23, 73, s)-Nets in Base 32
(73−23, 73, 2981)-Net over F32 — Constructive and digital
Digital (50, 73, 2981)-net over F32, using
- 321 times duplication [i] based on digital (49, 72, 2981)-net over F32, using
- net defined by OOA [i] based on linear OOA(3272, 2981, F32, 23, 23) (dual of [(2981, 23), 68491, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3272, 32792, F32, 23) (dual of [32792, 32720, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(3267, 32769, F32, 23) (dual of [32769, 32702, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,11]) ⊂ C([0,8]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(3272, 32792, F32, 23) (dual of [32792, 32720, 24]-code), using
- net defined by OOA [i] based on linear OOA(3272, 2981, F32, 23, 23) (dual of [(2981, 23), 68491, 24]-NRT-code), using
(73−23, 73, 5958)-Net in Base 32 — Constructive
(50, 73, 5958)-net in base 32, using
- net defined by OOA [i] based on OOA(3273, 5958, S32, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3273, 65539, S32, 23), using
- 1 times code embedding in larger space [i] based on OA(3272, 65538, S32, 23), using
- discarding parts of the base [i] based on linear OA(25645, 65538, F256, 23) (dual of [65538, 65493, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(25645, 65538, F256, 23) (dual of [65538, 65493, 24]-code), using
- 1 times code embedding in larger space [i] based on OA(3272, 65538, S32, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3273, 65539, S32, 23), using
(73−23, 73, 32795)-Net over F32 — Digital
Digital (50, 73, 32795)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3273, 32795, F32, 23) (dual of [32795, 32722, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(15) [i] based on
- linear OA(3267, 32768, F32, 23) (dual of [32768, 32701, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(22) ⊂ Ce(15) [i] based on
(73−23, 73, large)-Net in Base 32 — Upper bound on s
There is no (50, 73, large)-net in base 32, because
- 21 times m-reduction [i] would yield (50, 52, large)-net in base 32, but