Best Known (65, 65+17, s)-Nets in Base 5
(65, 65+17, 1954)-Net over F5 — Constructive and digital
Digital (65, 82, 1954)-net over F5, using
- 51 times duplication [i] based on digital (64, 81, 1954)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 1954, F5, 17, 17) (dual of [(1954, 17), 33137, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(581, 15633, F5, 17) (dual of [15633, 15552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 15634, F5, 17) (dual of [15634, 15553, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 15634, F5, 17) (dual of [15634, 15553, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(581, 15633, F5, 17) (dual of [15633, 15552, 18]-code), using
- net defined by OOA [i] based on linear OOA(581, 1954, F5, 17, 17) (dual of [(1954, 17), 33137, 18]-NRT-code), using
(65, 65+17, 9543)-Net over F5 — Digital
Digital (65, 82, 9543)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(582, 9543, F5, 17) (dual of [9543, 9461, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(582, 15640, F5, 17) (dual of [15640, 15558, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(582, 15640, F5, 17) (dual of [15640, 15558, 18]-code), using
(65, 65+17, large)-Net in Base 5 — Upper bound on s
There is no (65, 82, large)-net in base 5, because
- 15 times m-reduction [i] would yield (65, 67, large)-net in base 5, but