Best Known (56, 56+13, s)-Nets in Base 5
(56, 56+13, 2619)-Net over F5 — Constructive and digital
Digital (56, 69, 2619)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 14)-net over F5, using
- 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
- net defined by OOA [i] based on linear OOA(561, 2605, F5, 13, 13) (dual of [(2605, 13), 33804, 14]-NRT-code), using
(56, 56+13, 15659)-Net over F5 — Digital
Digital (56, 69, 15659)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 15659, F5, 13) (dual of [15659, 15590, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(567, 15655, F5, 13) (dual of [15655, 15588, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [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(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(567, 15657, F5, 12) (dual of [15657, 15590, 13]-code), using Gilbert–Varšamov bound and bm = 567 > Vbs−1(k−1) = 1450 585839 659587 708339 692858 731465 423762 358625 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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
- linear OA(567, 15655, F5, 13) (dual of [15655, 15588, 14]-code), using
- construction X with Varšamov bound [i] based on
(56, 56+13, large)-Net in Base 5 — Upper bound on s
There is no (56, 69, large)-net in base 5, because
- 11 times m-reduction [i] would yield (56, 58, large)-net in base 5, but