Best Known (53, 53+7, s)-Nets in Base 5
(53, 53+7, 2800900)-Net over F5 — Constructive and digital
Digital (53, 60, 2800900)-net over F5, using
- net defined by OOA [i] based on linear OOA(560, 2800900, F5, 9, 7) (dual of [(2800900, 9), 25208040, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(560, 2800901, F5, 3, 7) (dual of [(2800901, 3), 8402643, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(59, 4700, F5, 3, 3) (dual of [(4700, 3), 14091, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(59, 4700, F5, 2, 3) (dual of [(4700, 2), 9391, 4]-NRT-code), using
- linear OOA(551, 2796201, F5, 3, 7) (dual of [(2796201, 3), 8388552, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OOA 3-folding [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- linear OOA(59, 4700, F5, 3, 3) (dual of [(4700, 3), 14091, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(560, 2800901, F5, 3, 7) (dual of [(2800901, 3), 8402643, 8]-NRT-code), using
(53, 53+7, large)-Net over F5 — Digital
Digital (53, 60, large)-net over F5, using
- 58 times duplication [i] based on digital (45, 52, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
(53, 53+7, large)-Net in Base 5 — Upper bound on s
There is no (53, 60, large)-net in base 5, because
- 5 times m-reduction [i] would yield (53, 55, large)-net in base 5, but