Best Known (81−20, 81, s)-Nets in Base 25
(81−20, 81, 39064)-Net over F25 — Constructive and digital
Digital (61, 81, 39064)-net over F25, using
- 251 times duplication [i] based on digital (60, 80, 39064)-net over F25, using
- net defined by OOA [i] based on linear OOA(2580, 39064, F25, 20, 20) (dual of [(39064, 20), 781200, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2580, 390640, F25, 20) (dual of [390640, 390560, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2580, 390644, F25, 20) (dual of [390644, 390564, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(2577, 390625, F25, 20) (dual of [390625, 390548, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2561, 390625, F25, 16) (dual of [390625, 390564, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(253, 19, F25, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,25) or 19-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 Ce(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2580, 390644, F25, 20) (dual of [390644, 390564, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2580, 390640, F25, 20) (dual of [390640, 390560, 21]-code), using
- net defined by OOA [i] based on linear OOA(2580, 39064, F25, 20, 20) (dual of [(39064, 20), 781200, 21]-NRT-code), using
(81−20, 81, 390649)-Net over F25 — Digital
Digital (61, 81, 390649)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2581, 390649, F25, 20) (dual of [390649, 390568, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(2577, 390625, F25, 20) (dual of [390625, 390548, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2557, 390625, F25, 15) (dual of [390625, 390568, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-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(19) ⊂ Ce(14) [i] based on
(81−20, 81, large)-Net in Base 25 — Upper bound on s
There is no (61, 81, large)-net in base 25, because
- 18 times m-reduction [i] would yield (61, 63, large)-net in base 25, but