Best Known (52, 52+14, s)-Nets in Base 5
(52, 52+14, 558)-Net over F5 — Constructive and digital
Digital (52, 66, 558)-net over F5, using
- net defined by OOA [i] based on linear OOA(566, 558, F5, 14, 14) (dual of [(558, 14), 7746, 15]-NRT-code), using
(52, 52+14, 5025)-Net over F5 — Digital
Digital (52, 66, 5025)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(566, 5025, F5, 14) (dual of [5025, 4959, 15]-code), using
- 1885 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 29 times 0, 1, 66 times 0, 1, 134 times 0, 1, 236 times 0, 1, 359 times 0, 1, 472 times 0, 1, 566 times 0) [i] based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(50, 5, F5, 0) (dual of [5, 5, 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(13) ⊂ Ce(12) [i] based on
- 1885 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 29 times 0, 1, 66 times 0, 1, 134 times 0, 1, 236 times 0, 1, 359 times 0, 1, 472 times 0, 1, 566 times 0) [i] based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
(52, 52+14, 3289626)-Net in Base 5 — Upper bound on s
There is no (52, 66, 3289627)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 13552 530555 283739 178884 034301 375694 287129 023125 > 566 [i]