Best Known (86, 86+13, s)-Nets in Base 5
(86, 86+13, 325536)-Net over F5 — Constructive and digital
Digital (86, 99, 325536)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 14)-net over F5, using
- digital (78, 91, 325522)-net over F5, using
- net defined by OOA [i] based on linear OOA(591, 325522, F5, 13, 13) (dual of [(325522, 13), 4231695, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(591, 1953133, F5, 13) (dual of [1953133, 1953042, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(591, 1953134, F5, 13) (dual of [1953134, 1953043, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(591, 1953125, F5, 13) (dual of [1953125, 1953034, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(591, 1953134, F5, 13) (dual of [1953134, 1953043, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(591, 1953133, F5, 13) (dual of [1953133, 1953042, 14]-code), using
- net defined by OOA [i] based on linear OOA(591, 325522, F5, 13, 13) (dual of [(325522, 13), 4231695, 14]-NRT-code), using
(86, 86+13, 1953170)-Net over F5 — Digital
Digital (86, 99, 1953170)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(599, 1953170, F5, 13) (dual of [1953170, 1953071, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(598, 1953168, F5, 13) (dual of [1953168, 1953070, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(591, 1953125, F5, 13) (dual of [1953125, 1953034, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(57, 43, F5, 4) (dual of [43, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(598, 1953169, F5, 12) (dual of [1953169, 1953071, 13]-code), using Gilbert–Varšamov bound and bm = 598 > Vbs−1(k−1) = 165 816663 570414 808036 081693 373320 996736 819120 813401 943573 507869 931969 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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(598, 1953168, F5, 13) (dual of [1953168, 1953070, 14]-code), using
- construction X with Varšamov bound [i] based on
(86, 86+13, large)-Net in Base 5 — Upper bound on s
There is no (86, 99, large)-net in base 5, because
- 11 times m-reduction [i] would yield (86, 88, large)-net in base 5, but