Best Known (33, 33+7, s)-Nets in Base 7
(33, 33+7, 78438)-Net over F7 — Constructive and digital
Digital (33, 40, 78438)-net over F7, using
- net defined by OOA [i] based on linear OOA(740, 78438, F7, 7, 7) (dual of [(78438, 7), 549026, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(740, 235315, F7, 7) (dual of [235315, 235275, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(740, 235316, F7, 7) (dual of [235316, 235276, 8]-code), using
- trace code [i] based on linear OA(4920, 117658, F49, 7) (dual of [117658, 117638, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(4919, 117650, F49, 7) (dual of [117650, 117631, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(4913, 117650, F49, 5) (dual of [117650, 117637, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(497, 8, F49, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,49)), using
- dual of repetition code with length 8 [i]
- linear OA(491, 8, F49, 1) (dual of [8, 7, 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
- trace code [i] based on linear OA(4920, 117658, F49, 7) (dual of [117658, 117638, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(740, 235316, F7, 7) (dual of [235316, 235276, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(740, 235315, F7, 7) (dual of [235315, 235275, 8]-code), using
(33, 33+7, 235316)-Net over F7 — Digital
Digital (33, 40, 235316)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(740, 235316, F7, 7) (dual of [235316, 235276, 8]-code), using
- trace code [i] based on linear OA(4920, 117658, F49, 7) (dual of [117658, 117638, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(4919, 117650, F49, 7) (dual of [117650, 117631, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(4913, 117650, F49, 5) (dual of [117650, 117637, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(497, 8, F49, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,49)), using
- dual of repetition code with length 8 [i]
- linear OA(491, 8, F49, 1) (dual of [8, 7, 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
- trace code [i] based on linear OA(4920, 117658, F49, 7) (dual of [117658, 117638, 8]-code), using
(33, 33+7, large)-Net in Base 7 — Upper bound on s
There is no (33, 40, large)-net in base 7, because
- 5 times m-reduction [i] would yield (33, 35, large)-net in base 7, but