Best Known (48, 61, s)-Nets in Base 5
(48, 61, 2605)-Net over F5 — Constructive and digital
Digital (48, 61, 2605)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 2605, F5, 13, 13) (dual of [(2605, 13), 33804, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(561, 15631, F5, 13) (dual of [15631, 15570, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(561, 15631, F5, 13) (dual of [15631, 15570, 14]-code), using
(48, 61, 7964)-Net over F5 — Digital
Digital (48, 61, 7964)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 7964, F5, 13) (dual of [7964, 7903, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using
(48, 61, 7309066)-Net in Base 5 — Upper bound on s
There is no (48, 61, 7309067)-net in base 5, because
- 1 times m-reduction [i] would yield (48, 60, 7309067)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 867362 287428 962045 598046 068390 372889 486313 > 560 [i]