Best Known (41, 41+16, s)-Nets in Base 7
(41, 41+16, 302)-Net over F7 — Constructive and digital
Digital (41, 57, 302)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 302, F7, 16, 16) (dual of [(302, 16), 4775, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(757, 2416, F7, 16) (dual of [2416, 2359, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 2417, F7, 16) (dual of [2417, 2360, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(741, 2401, F7, 12) (dual of [2401, 2360, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(74, 16, F7, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,7)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(757, 2417, F7, 16) (dual of [2417, 2360, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(757, 2416, F7, 16) (dual of [2416, 2359, 17]-code), using
(41, 41+16, 2411)-Net over F7 — Digital
Digital (41, 57, 2411)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(757, 2411, F7, 16) (dual of [2411, 2354, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 2417, F7, 16) (dual of [2417, 2360, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(741, 2401, F7, 12) (dual of [2401, 2360, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(74, 16, F7, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,7)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(757, 2417, F7, 16) (dual of [2417, 2360, 17]-code), using
(41, 41+16, 658960)-Net in Base 7 — Upper bound on s
There is no (41, 57, 658961)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1 481128 229349 099100 513575 132393 595294 595965 108417 > 757 [i]