Best Known (57−18, 57, s)-Nets in Base 25
(57−18, 57, 1738)-Net over F25 — Constructive and digital
Digital (39, 57, 1738)-net over F25, using
- 251 times duplication [i] based on digital (38, 56, 1738)-net over F25, using
- net defined by OOA [i] based on linear OOA(2556, 1738, F25, 18, 18) (dual of [(1738, 18), 31228, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2556, 15642, F25, 18) (dual of [15642, 15586, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2556, 15644, F25, 18) (dual of [15644, 15588, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2556, 15644, F25, 18) (dual of [15644, 15588, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2556, 15642, F25, 18) (dual of [15642, 15586, 19]-code), using
- net defined by OOA [i] based on linear OOA(2556, 1738, F25, 18, 18) (dual of [(1738, 18), 31228, 19]-NRT-code), using
(57−18, 57, 15648)-Net over F25 — Digital
Digital (39, 57, 15648)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2557, 15648, F25, 18) (dual of [15648, 15591, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2534, 15625, F25, 12) (dual of [15625, 15591, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(255, 23, F25, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- Reed–Solomon code RS(20,25) [i]
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
(57−18, 57, large)-Net in Base 25 — Upper bound on s
There is no (39, 57, large)-net in base 25, because
- 16 times m-reduction [i] would yield (39, 41, large)-net in base 25, but