Best Known (60−8, 60, s)-Nets in Base 5
(60−8, 60, 488295)-Net over F5 — Constructive and digital
Digital (52, 60, 488295)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net over F5, using
- digital (47, 55, 488283)-net over F5, using
- net defined by OOA [i] based on linear OOA(555, 488283, F5, 8, 8) (dual of [(488283, 8), 3906209, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(555, 1953132, F5, 8) (dual of [1953132, 1953077, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(555, 1953134, F5, 8) (dual of [1953134, 1953079, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(555, 1953134, F5, 8) (dual of [1953134, 1953079, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(555, 1953132, F5, 8) (dual of [1953132, 1953077, 9]-code), using
- net defined by OOA [i] based on linear OOA(555, 488283, F5, 8, 8) (dual of [(488283, 8), 3906209, 9]-NRT-code), using
(60−8, 60, 1953157)-Net over F5 — Digital
Digital (52, 60, 1953157)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(560, 1953157, F5, 8) (dual of [1953157, 1953097, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(60−8, 60, large)-Net in Base 5 — Upper bound on s
There is no (52, 60, large)-net in base 5, because
- 6 times m-reduction [i] would yield (52, 54, large)-net in base 5, but