Best Known (40, 52, s)-Nets in Base 7
(40, 52, 2803)-Net over F7 — Constructive and digital
Digital (40, 52, 2803)-net over F7, using
- net defined by OOA [i] based on linear OOA(752, 2803, F7, 12, 12) (dual of [(2803, 12), 33584, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(752, 16818, F7, 12) (dual of [16818, 16766, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(752, 16818, F7, 12) (dual of [16818, 16766, 13]-code), using
(40, 52, 15405)-Net over F7 — Digital
Digital (40, 52, 15405)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(752, 15405, F7, 12) (dual of [15405, 15353, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(752, 16818, F7, 12) (dual of [16818, 16766, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(752, 16818, F7, 12) (dual of [16818, 16766, 13]-code), using
(40, 52, large)-Net in Base 7 — Upper bound on s
There is no (40, 52, large)-net in base 7, because
- 10 times m-reduction [i] would yield (40, 42, large)-net in base 7, but