Best Known (20−10, 20, s)-Nets in Base 32
(20−10, 20, 206)-Net over F32 — Constructive and digital
Digital (10, 20, 206)-net over F32, using
- net defined by OOA [i] based on linear OOA(3220, 206, F32, 10, 10) (dual of [(206, 10), 2040, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3220, 1030, F32, 10) (dual of [1030, 1010, 11]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,7]) [i] based on
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3219, 1023, F32, 10) (dual of [1023, 1004, 11]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,7}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,7]) [i] based on
- OA 5-folding and stacking [i] based on linear OA(3220, 1030, F32, 10) (dual of [1030, 1010, 11]-code), using
(20−10, 20, 259)-Net in Base 32 — Constructive
(10, 20, 259)-net in base 32, using
- 1 times m-reduction [i] based on (10, 21, 259)-net in base 32, using
- base change [i] based on (4, 15, 259)-net in base 128, using
- 1 times m-reduction [i] based on (4, 16, 259)-net in base 128, using
- base change [i] based on digital (2, 14, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 14, 259)-net over F256, using
- 1 times m-reduction [i] based on (4, 16, 259)-net in base 128, using
- base change [i] based on (4, 15, 259)-net in base 128, using
(20−10, 20, 515)-Net over F32 — Digital
Digital (10, 20, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3220, 515, F32, 2, 10) (dual of [(515, 2), 1010, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3220, 1030, F32, 10) (dual of [1030, 1010, 11]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,7]) [i] based on
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3219, 1023, F32, 10) (dual of [1023, 1004, 11]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,7}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,7]) [i] based on
- OOA 2-folding [i] based on linear OA(3220, 1030, F32, 10) (dual of [1030, 1010, 11]-code), using
(20−10, 20, 88117)-Net in Base 32 — Upper bound on s
There is no (10, 20, 88118)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1 267662 489714 718757 483392 191755 > 3220 [i]