Best Known (61, 61+9, s)-Nets in Base 5
(61, 61+9, 488310)-Net over F5 — Constructive and digital
Digital (61, 70, 488310)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 27)-net over F5, using
- digital (55, 64, 488283)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 488283, F5, 9, 9) (dual of [(488283, 9), 4394483, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(564, 1953133, F5, 9) (dual of [1953133, 1953069, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(564, 1953134, F5, 9) (dual of [1953134, 1953070, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 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(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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(564, 1953134, F5, 9) (dual of [1953134, 1953070, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(564, 1953133, F5, 9) (dual of [1953133, 1953069, 10]-code), using
- net defined by OOA [i] based on linear OOA(564, 488283, F5, 9, 9) (dual of [(488283, 9), 4394483, 10]-NRT-code), using
(61, 61+9, 1953164)-Net over F5 — Digital
Digital (61, 70, 1953164)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(570, 1953164, F5, 9) (dual of [1953164, 1953094, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- linear OA(564, 1953134, F5, 9) (dual of [1953134, 1953070, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 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(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(8) ⊂ Ce(7) [i] based on
- (u, u+v)-construction [i] based on
(61, 61+9, large)-Net in Base 5 — Upper bound on s
There is no (61, 70, large)-net in base 5, because
- 7 times m-reduction [i] would yield (61, 63, large)-net in base 5, but