Best Known (26, 26+13, s)-Nets in Base 25
(26, 26+13, 2605)-Net over F25 — Constructive and digital
Digital (26, 39, 2605)-net over F25, using
- 251 times duplication [i] based on digital (25, 38, 2605)-net over F25, using
- net defined by OOA [i] based on linear OOA(2538, 2605, F25, 13, 13) (dual of [(2605, 13), 33827, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2538, 15631, F25, 13) (dual of [15631, 15593, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2538, 15633, F25, 13) (dual of [15633, 15595, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(2537, 15626, F25, 13) (dual of [15626, 15589, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2538, 15633, F25, 13) (dual of [15633, 15595, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2538, 15631, F25, 13) (dual of [15631, 15593, 14]-code), using
- net defined by OOA [i] based on linear OOA(2538, 2605, F25, 13, 13) (dual of [(2605, 13), 33827, 14]-NRT-code), using
(26, 26+13, 13800)-Net over F25 — Digital
Digital (26, 39, 13800)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2539, 13800, F25, 13) (dual of [13800, 13761, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2539, 15636, F25, 13) (dual of [15636, 15597, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(2537, 15625, F25, 13) (dual of [15625, 15588, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(2539, 15636, F25, 13) (dual of [15636, 15597, 14]-code), using
(26, 26+13, large)-Net in Base 25 — Upper bound on s
There is no (26, 39, large)-net in base 25, because
- 11 times m-reduction [i] would yield (26, 28, large)-net in base 25, but