Best Known (47, 63, s)-Nets in Base 5
(47, 63, 391)-Net over F5 — Constructive and digital
Digital (47, 63, 391)-net over F5, using
- 51 times duplication [i] based on digital (46, 62, 391)-net over F5, using
- net defined by OOA [i] based on linear OOA(562, 391, F5, 16, 16) (dual of [(391, 16), 6194, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(562, 3128, F5, 16) (dual of [3128, 3066, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(562, 3131, F5, 16) (dual of [3131, 3069, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(562, 3131, F5, 16) (dual of [3131, 3069, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(562, 3128, F5, 16) (dual of [3128, 3066, 17]-code), using
- net defined by OOA [i] based on linear OOA(562, 391, F5, 16, 16) (dual of [(391, 16), 6194, 17]-NRT-code), using
(47, 63, 1874)-Net over F5 — Digital
Digital (47, 63, 1874)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(563, 1874, F5, 16) (dual of [1874, 1811, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(563, 3133, F5, 16) (dual of [3133, 3070, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(563, 3133, F5, 16) (dual of [3133, 3070, 17]-code), using
(47, 63, 300615)-Net in Base 5 — Upper bound on s
There is no (47, 63, 300616)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 108 422572 243709 447922 166757 296731 220900 752385 > 563 [i]