Best Known (50−12, 50, s)-Nets in Base 5
(50−12, 50, 524)-Net over F5 — Constructive and digital
Digital (38, 50, 524)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 524, F5, 12, 12) (dual of [(524, 12), 6238, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(550, 3144, F5, 12) (dual of [3144, 3094, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(550, 3144, F5, 12) (dual of [3144, 3094, 13]-code), using
(50−12, 50, 3006)-Net over F5 — Digital
Digital (38, 50, 3006)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(550, 3006, F5, 12) (dual of [3006, 2956, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 3144, F5, 12) (dual of [3144, 3094, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 3144, F5, 12) (dual of [3144, 3094, 13]-code), using
(50−12, 50, 499929)-Net in Base 5 — Upper bound on s
There is no (38, 50, 499930)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 88818 512103 696311 831307 428571 944065 > 550 [i]