Best Known (32, 32+15, s)-Nets in Base 25
(32, 32+15, 2234)-Net over F25 — Constructive and digital
Digital (32, 47, 2234)-net over F25, using
- 251 times duplication [i] based on digital (31, 46, 2234)-net over F25, using
- net defined by OOA [i] based on linear OOA(2546, 2234, F25, 15, 15) (dual of [(2234, 15), 33464, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2546, 15639, F25, 15) (dual of [15639, 15593, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2546, 15641, F25, 15) (dual of [15641, 15595, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(2543, 15626, F25, 15) (dual of [15626, 15583, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2531, 15626, F25, 11) (dual of [15626, 15595, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(253, 15, F25, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2546, 15641, F25, 15) (dual of [15641, 15595, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2546, 15639, F25, 15) (dual of [15639, 15593, 16]-code), using
- net defined by OOA [i] based on linear OOA(2546, 2234, F25, 15, 15) (dual of [(2234, 15), 33464, 16]-NRT-code), using
(32, 32+15, 15644)-Net over F25 — Digital
Digital (32, 47, 15644)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2547, 15644, F25, 15) (dual of [15644, 15597, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(2543, 15625, F25, 15) (dual of [15625, 15582, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2528, 15625, F25, 10) (dual of [15625, 15597, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(254, 19, F25, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
(32, 32+15, large)-Net in Base 25 — Upper bound on s
There is no (32, 47, large)-net in base 25, because
- 13 times m-reduction [i] would yield (32, 34, large)-net in base 25, but