Best Known (19, 19+9, s)-Nets in Base 32
(19, 19+9, 8195)-Net over F32 — Constructive and digital
Digital (19, 28, 8195)-net over F32, using
- net defined by OOA [i] based on linear OOA(3228, 8195, F32, 9, 9) (dual of [(8195, 9), 73727, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3228, 32781, F32, 9) (dual of [32781, 32753, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 32784, F32, 9) (dual of [32784, 32756, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(3213, 32769, F32, 5) (dual of [32769, 32756, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 32784, F32, 9) (dual of [32784, 32756, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3228, 32781, F32, 9) (dual of [32781, 32753, 10]-code), using
(19, 19+9, 16384)-Net in Base 32 — Constructive
(19, 28, 16384)-net in base 32, using
- net defined by OOA [i] based on OOA(3228, 16384, S32, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(3228, 65537, S32, 9), using
- discarding factors based on OA(3228, 65538, S32, 9), using
- discarding parts of the base [i] based on linear OA(25617, 65538, F256, 9) (dual of [65538, 65521, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(25617, 65538, F256, 9) (dual of [65538, 65521, 10]-code), using
- discarding factors based on OA(3228, 65538, S32, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(3228, 65537, S32, 9), using
(19, 19+9, 32784)-Net over F32 — Digital
Digital (19, 28, 32784)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3228, 32784, F32, 9) (dual of [32784, 32756, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(3213, 32769, F32, 5) (dual of [32769, 32756, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
(19, 19+9, large)-Net in Base 32 — Upper bound on s
There is no (19, 28, large)-net in base 32, because
- 7 times m-reduction [i] would yield (19, 21, large)-net in base 32, but