Best Known (51, 51+7, s)-Nets in Base 5
(51, 51+7, 2796875)-Net over F5 — Constructive and digital
Digital (51, 58, 2796875)-net over F5, using
- net defined by OOA [i] based on linear OOA(558, 2796875, F5, 9, 7) (dual of [(2796875, 9), 25171817, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(558, 2796876, F5, 3, 7) (dual of [(2796876, 3), 8390570, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(57, 675, F5, 3, 3) (dual of [(675, 3), 2018, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(57, 675, F5, 2, 3) (dual of [(675, 2), 1343, 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(57, 675, F5, 3, 3) (dual of [(675, 3), 2018, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(558, 2796876, F5, 3, 7) (dual of [(2796876, 3), 8390570, 8]-NRT-code), using
(51, 51+7, large)-Net over F5 — Digital
Digital (51, 58, large)-net over F5, using
- 56 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
(51, 51+7, large)-Net in Base 5 — Upper bound on s
There is no (51, 58, large)-net in base 5, because
- 5 times m-reduction [i] would yield (51, 53, large)-net in base 5, but