Best Known (54, 54+9, s)-Nets in Base 5
(54, 54+9, 97685)-Net over F5 — Constructive and digital
Digital (54, 63, 97685)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 27)-net over F5, using
- digital (48, 57, 97658)-net over F5, using
- net defined by OOA [i] based on linear OOA(557, 97658, F5, 9, 9) (dual of [(97658, 9), 878865, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(557, 390633, F5, 9) (dual of [390633, 390576, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(557, 390633, F5, 9) (dual of [390633, 390576, 10]-code), using
- net defined by OOA [i] based on linear OOA(557, 97658, F5, 9, 9) (dual of [(97658, 9), 878865, 10]-NRT-code), using
(54, 54+9, 390663)-Net over F5 — Digital
Digital (54, 63, 390663)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(563, 390663, F5, 9) (dual of [390663, 390600, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- linear OA(557, 390633, F5, 9) (dual of [390633, 390576, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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(8) ⊂ Ce(7) [i] based on
- (u, u+v)-construction [i] based on
(54, 54+9, large)-Net in Base 5 — Upper bound on s
There is no (54, 63, large)-net in base 5, because
- 7 times m-reduction [i] would yield (54, 56, large)-net in base 5, but