Best Known (58, 58+7, s)-Nets in Base 7
(58, 58+7, 7686415)-Net over F7 — Constructive and digital
Digital (58, 65, 7686415)-net over F7, using
- net defined by OOA [i] based on linear OOA(765, 7686415, F7, 9, 7) (dual of [(7686415, 9), 69177670, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(765, 7686416, F7, 3, 7) (dual of [(7686416, 3), 23059183, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(713, 6120101, F7, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(713, 6120101, F7, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(713, 6120101, F7, 3) (dual of [6120101, 6120088, 4]-code or 6120101-cap in PG(12,7)), using
- appending kth column [i] based on linear OOA(713, 6120101, F7, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
- linear OOA(752, 3843208, F7, 3, 7) (dual of [(3843208, 3), 11529572, 8]-NRT-code), using
- trace code [i] based on linear OOA(4926, 1921604, F49, 3, 7) (dual of [(1921604, 3), 5764786, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4926, 5764812, F49, 7) (dual of [5764812, 5764786, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(4925, 5764802, F49, 7) (dual of [5764802, 5764777, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(4917, 5764802, F49, 5) (dual of [5764802, 5764785, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(499, 10, F49, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,49)), using
- dual of repetition code with length 10 [i]
- linear OA(491, 10, F49, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- Reed–Solomon code RS(48,49) [i]
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding [i] based on linear OA(4926, 5764812, F49, 7) (dual of [5764812, 5764786, 8]-code), using
- trace code [i] based on linear OOA(4926, 1921604, F49, 3, 7) (dual of [(1921604, 3), 5764786, 8]-NRT-code), using
- linear OOA(713, 6120101, F7, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(765, 7686416, F7, 3, 7) (dual of [(7686416, 3), 23059183, 8]-NRT-code), using
(58, 58+7, large)-Net over F7 — Digital
Digital (58, 65, large)-net over F7, using
- 71 times duplication [i] based on digital (57, 64, large)-net over F7, using
- t-expansion [i] based on digital (55, 64, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- t-expansion [i] based on digital (55, 64, large)-net over F7, using
(58, 58+7, large)-Net in Base 7 — Upper bound on s
There is no (58, 65, large)-net in base 7, because
- 5 times m-reduction [i] would yield (58, 60, large)-net in base 7, but