Best Known (67, 67+14, s)-Nets in Base 7
(67, 67+14, 33615)-Net over F7 — Constructive and digital
Digital (67, 81, 33615)-net over F7, using
- net defined by OOA [i] based on linear OOA(781, 33615, F7, 14, 14) (dual of [(33615, 14), 470529, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(781, 235305, F7, 14) (dual of [235305, 235224, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(780, 235304, F7, 14) (dual of [235304, 235224, 15]-code), using
- trace code [i] based on linear OA(4940, 117652, F49, 14) (dual of [117652, 117612, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(4940, 117652, F49, 14) (dual of [117652, 117612, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(780, 235304, F7, 14) (dual of [235304, 235224, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(781, 235305, F7, 14) (dual of [235305, 235224, 15]-code), using
(67, 67+14, 235306)-Net over F7 — Digital
Digital (67, 81, 235306)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(781, 235306, F7, 14) (dual of [235306, 235225, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(780, 235304, F7, 14) (dual of [235304, 235224, 15]-code), using
- trace code [i] based on linear OA(4940, 117652, F49, 14) (dual of [117652, 117612, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(4940, 117652, F49, 14) (dual of [117652, 117612, 15]-code), using
- linear OA(780, 235305, F7, 13) (dual of [235305, 235225, 14]-code), using Gilbert–Varšamov bound and bm = 780 > Vbs−1(k−1) = 130891 557296 519769 839785 466292 947424 896454 705716 642724 371031 775681 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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
- linear OA(780, 235304, F7, 14) (dual of [235304, 235224, 15]-code), using
- construction X with Varšamov bound [i] based on
(67, 67+14, large)-Net in Base 7 — Upper bound on s
There is no (67, 81, large)-net in base 7, because
- 12 times m-reduction [i] would yield (67, 69, large)-net in base 7, but