Best Known (37, 37+12, s)-Nets in Base 25
(37, 37+12, 65108)-Net over F25 — Constructive and digital
Digital (37, 49, 65108)-net over F25, using
- net defined by OOA [i] based on linear OOA(2549, 65108, F25, 12, 12) (dual of [(65108, 12), 781247, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2549, 390648, F25, 12) (dual of [390648, 390599, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2549, 390649, F25, 12) (dual of [390649, 390600, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(2545, 390625, F25, 12) (dual of [390625, 390580, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2525, 390625, F25, 7) (dual of [390625, 390600, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(11) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(2549, 390649, F25, 12) (dual of [390649, 390600, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2549, 390648, F25, 12) (dual of [390648, 390599, 13]-code), using
(37, 37+12, 390649)-Net over F25 — Digital
Digital (37, 49, 390649)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2549, 390649, F25, 12) (dual of [390649, 390600, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(2545, 390625, F25, 12) (dual of [390625, 390580, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2525, 390625, F25, 7) (dual of [390625, 390600, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(11) ⊂ Ce(6) [i] based on
(37, 37+12, large)-Net in Base 25 — Upper bound on s
There is no (37, 49, large)-net in base 25, because
- 10 times m-reduction [i] would yield (37, 39, large)-net in base 25, but