Best Known (54−16, 54, s)-Nets in Base 7
(54−16, 54, 300)-Net over F7 — Constructive and digital
Digital (38, 54, 300)-net over F7, using
- 71 times duplication [i] based on digital (37, 53, 300)-net over F7, using
- net defined by OOA [i] based on linear OOA(753, 300, F7, 16, 16) (dual of [(300, 16), 4747, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(753, 2400, F7, 16) (dual of [2400, 2347, 17]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(753, 2400, F7, 16) (dual of [2400, 2347, 17]-code), using
- net defined by OOA [i] based on linear OOA(753, 300, F7, 16, 16) (dual of [(300, 16), 4747, 17]-NRT-code), using
(54−16, 54, 1586)-Net over F7 — Digital
Digital (38, 54, 1586)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(754, 1586, F7, 16) (dual of [1586, 1532, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(754, 2406, F7, 16) (dual of [2406, 2352, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [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(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(754, 2406, F7, 16) (dual of [2406, 2352, 17]-code), using
(54−16, 54, 317646)-Net in Base 7 — Upper bound on s
There is no (38, 54, 317647)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 4318 176693 804142 736472 509228 108955 071628 325585 > 754 [i]