Best Known (60, 60+29, s)-Nets in Base 25
(60, 60+29, 1117)-Net over F25 — Constructive and digital
Digital (60, 89, 1117)-net over F25, using
- 253 times duplication [i] based on digital (57, 86, 1117)-net over F25, using
- net defined by OOA [i] based on linear OOA(2586, 1117, F25, 29, 29) (dual of [(1117, 29), 32307, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2586, 15639, F25, 29) (dual of [15639, 15553, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2586, 15641, F25, 29) (dual of [15641, 15555, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2570, 15625, F25, 24) (dual of [15625, 15555, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(254, 16, F25, 4) (dual of [16, 12, 5]-code or 16-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(28) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(2586, 15641, F25, 29) (dual of [15641, 15555, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2586, 15639, F25, 29) (dual of [15639, 15553, 30]-code), using
- net defined by OOA [i] based on linear OOA(2586, 1117, F25, 29, 29) (dual of [(1117, 29), 32307, 30]-NRT-code), using
(60, 60+29, 15651)-Net over F25 — Digital
Digital (60, 89, 15651)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2589, 15651, F25, 29) (dual of [15651, 15562, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2561, 15625, F25, 21) (dual of [15625, 15564, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(257, 26, F25, 7) (dual of [26, 19, 8]-code or 26-arc in PG(6,25)), using
- extended Reed–Solomon code RSe(19,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- algebraic-geometric code AG(F,9P) with degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F,6P) with degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
(60, 60+29, large)-Net in Base 25 — Upper bound on s
There is no (60, 89, large)-net in base 25, because
- 27 times m-reduction [i] would yield (60, 62, large)-net in base 25, but