Best Known (54, 54+27, s)-Nets in Base 25
(54, 54+27, 1203)-Net over F25 — Constructive and digital
Digital (54, 81, 1203)-net over F25, using
- 251 times duplication [i] based on digital (53, 80, 1203)-net over F25, using
- net defined by OOA [i] based on linear OOA(2580, 1203, F25, 27, 27) (dual of [(1203, 27), 32401, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2580, 15640, F25, 27) (dual of [15640, 15560, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2580, 15641, F25, 27) (dual of [15641, 15561, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(2576, 15625, F25, 27) (dual of [15625, 15549, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2564, 15625, F25, 22) (dual of [15625, 15561, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(2580, 15641, F25, 27) (dual of [15641, 15561, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2580, 15640, F25, 27) (dual of [15640, 15560, 28]-code), using
- net defined by OOA [i] based on linear OOA(2580, 1203, F25, 27, 27) (dual of [(1203, 27), 32401, 28]-NRT-code), using
(54, 54+27, 12601)-Net over F25 — Digital
Digital (54, 81, 12601)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2581, 12601, F25, 27) (dual of [12601, 12520, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2581, 15645, F25, 27) (dual of [15645, 15564, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(2576, 15625, F25, 27) (dual of [15625, 15549, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(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(26) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2581, 15645, F25, 27) (dual of [15645, 15564, 28]-code), using
(54, 54+27, large)-Net in Base 25 — Upper bound on s
There is no (54, 81, large)-net in base 25, because
- 25 times m-reduction [i] would yield (54, 56, large)-net in base 25, but