Best Known (66, 66+14, s)-Nets in Base 7
(66, 66+14, 33614)-Net over F7 — Constructive and digital
Digital (66, 80, 33614)-net over F7, using
- net defined by OOA [i] based on linear OOA(780, 33614, F7, 14, 14) (dual of [(33614, 14), 470516, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(780, 235298, F7, 14) (dual of [235298, 235218, 15]-code), using
- trace code [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]
- trace code [i] based on linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(780, 235298, F7, 14) (dual of [235298, 235218, 15]-code), using
(66, 66+14, 235304)-Net over F7 — Digital
Digital (66, 80, 235304)-net over F7, using
- embedding of OOA with Gilbert–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
(66, 66+14, large)-Net in Base 7 — Upper bound on s
There is no (66, 80, large)-net in base 7, because
- 12 times m-reduction [i] would yield (66, 68, large)-net in base 7, but