Best Known (52, 52+25, s)-Nets in Base 25
(52, 52+25, 1303)-Net over F25 — Constructive and digital
Digital (52, 77, 1303)-net over F25, using
- 251 times duplication [i] based on digital (51, 76, 1303)-net over F25, using
- net defined by OOA [i] based on linear OOA(2576, 1303, F25, 25, 25) (dual of [(1303, 25), 32499, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2576, 15637, F25, 25) (dual of [15637, 15561, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2576, 15641, F25, 25) (dual of [15641, 15565, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2573, 15626, F25, 25) (dual of [15626, 15553, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2561, 15626, F25, 21) (dual of [15626, 15565, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2576, 15641, F25, 25) (dual of [15641, 15565, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2576, 15637, F25, 25) (dual of [15637, 15561, 26]-code), using
- net defined by OOA [i] based on linear OOA(2576, 1303, F25, 25, 25) (dual of [(1303, 25), 32499, 26]-NRT-code), using
(52, 52+25, 15644)-Net over F25 — Digital
Digital (52, 77, 15644)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2577, 15644, F25, 25) (dual of [15644, 15567, 26]-code), using
- 1 times truncation [i] based on linear OA(2578, 15645, F25, 26) (dual of [15645, 15567, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(2573, 15625, F25, 26) (dual of [15625, 15552, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2558, 15625, F25, 20) (dual of [15625, 15567, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(255, 20, F25, 5) (dual of [20, 15, 6]-code or 20-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(25) ⊂ Ce(19) [i] based on
- 1 times truncation [i] based on linear OA(2578, 15645, F25, 26) (dual of [15645, 15567, 27]-code), using
(52, 52+25, large)-Net in Base 25 — Upper bound on s
There is no (52, 77, large)-net in base 25, because
- 23 times m-reduction [i] would yield (52, 54, large)-net in base 25, but