Best Known (48, 48+9, s)-Nets in Base 5
(48, 48+9, 97658)-Net over F5 — Constructive and digital
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
(48, 48+9, 330075)-Net over F5 — Digital
Digital (48, 57, 330075)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(557, 330075, F5, 9) (dual of [330075, 330018, 10]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using
(48, 48+9, large)-Net in Base 5 — Upper bound on s
There is no (48, 57, large)-net in base 5, because
- 7 times m-reduction [i] would yield (48, 50, large)-net in base 5, but