Best Known (53, 53+18, s)-Nets in Base 5
(53, 53+18, 347)-Net over F5 — Constructive and digital
Digital (53, 71, 347)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 347, F5, 18, 18) (dual of [(347, 18), 6175, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(571, 3123, F5, 18) (dual of [3123, 3052, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(571, 3123, F5, 18) (dual of [3123, 3052, 19]-code), using
(53, 53+18, 1932)-Net over F5 — Digital
Digital (53, 71, 1932)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(571, 1932, F5, 18) (dual of [1932, 1861, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using
(53, 53+18, 338672)-Net in Base 5 — Upper bound on s
There is no (53, 71, 338673)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 42 352026 221000 104192 816191 545531 579105 195030 720805 > 571 [i]