Best Known (76, 76+19, s)-Nets in Base 5
(76, 76+19, 1738)-Net over F5 — Constructive and digital
Digital (76, 95, 1738)-net over F5, using
- 51 times duplication [i] based on digital (75, 94, 1738)-net over F5, using
- net defined by OOA [i] based on linear OOA(594, 1738, F5, 19, 19) (dual of [(1738, 19), 32928, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(594, 15643, F5, 19) (dual of [15643, 15549, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(594, 15646, F5, 19) (dual of [15646, 15552, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(53, 21, F5, 2) (dual of [21, 18, 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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(594, 15646, F5, 19) (dual of [15646, 15552, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(594, 15643, F5, 19) (dual of [15643, 15549, 20]-code), using
- net defined by OOA [i] based on linear OOA(594, 1738, F5, 19, 19) (dual of [(1738, 19), 32928, 20]-NRT-code), using
(76, 76+19, 13134)-Net over F5 — Digital
Digital (76, 95, 13134)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(595, 13134, F5, 19) (dual of [13134, 13039, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(595, 15647, F5, 19) (dual of [15647, 15552, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(594, 15646, F5, 19) (dual of [15646, 15552, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(53, 21, F5, 2) (dual of [21, 18, 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(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(594, 15646, F5, 19) (dual of [15646, 15552, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(595, 15647, F5, 19) (dual of [15647, 15552, 20]-code), using
(76, 76+19, large)-Net in Base 5 — Upper bound on s
There is no (76, 95, large)-net in base 5, because
- 17 times m-reduction [i] would yield (76, 78, large)-net in base 5, but