Best Known (77−16, 77, s)-Nets in Base 5
(77−16, 77, 1955)-Net over F5 — Constructive and digital
Digital (61, 77, 1955)-net over F5, using
- 51 times duplication [i] based on digital (60, 76, 1955)-net over F5, using
- net defined by OOA [i] based on linear OOA(576, 1955, F5, 16, 16) (dual of [(1955, 16), 31204, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(576, 15640, F5, 16) (dual of [15640, 15564, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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(53, 15, F5, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OA 8-folding and stacking [i] based on linear OA(576, 15640, F5, 16) (dual of [15640, 15564, 17]-code), using
- net defined by OOA [i] based on linear OOA(576, 1955, F5, 16, 16) (dual of [(1955, 16), 31204, 17]-NRT-code), using
(77−16, 77, 9405)-Net over F5 — Digital
Digital (61, 77, 9405)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(577, 9405, F5, 16) (dual of [9405, 9328, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 15647, F5, 16) (dual of [15647, 15570, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(577, 15647, F5, 16) (dual of [15647, 15570, 17]-code), using
(77−16, 77, 5025923)-Net in Base 5 — Upper bound on s
There is no (61, 77, 5025924)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 661744 813111 829056 793225 110675 093990 803288 510414 486529 > 577 [i]