Best Known (27, 27+6, s)-Nets in Base 7
(27, 27+6, 78435)-Net over F7 — Constructive and digital
Digital (27, 33, 78435)-net over F7, using
- net defined by OOA [i] based on linear OOA(733, 78435, F7, 6, 6) (dual of [(78435, 6), 470577, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(733, 235305, F7, 6) (dual of [235305, 235272, 7]-code), using
- 1 times code embedding in larger space [i] based on linear OA(732, 235304, F7, 6) (dual of [235304, 235272, 7]-code), using
- trace code [i] based on linear OA(4916, 117652, F49, 6) (dual of [117652, 117636, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(4916, 117649, F49, 6) (dual of [117649, 117633, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(4913, 117649, F49, 5) (dual of [117649, 117636, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(4916, 117652, F49, 6) (dual of [117652, 117636, 7]-code), using
- 1 times code embedding in larger space [i] based on linear OA(732, 235304, F7, 6) (dual of [235304, 235272, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(733, 235305, F7, 6) (dual of [235305, 235272, 7]-code), using
(27, 27+6, 235306)-Net over F7 — Digital
Digital (27, 33, 235306)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(733, 235306, F7, 6) (dual of [235306, 235273, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(732, 235304, F7, 6) (dual of [235304, 235272, 7]-code), using
- trace code [i] based on linear OA(4916, 117652, F49, 6) (dual of [117652, 117636, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(4916, 117649, F49, 6) (dual of [117649, 117633, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(4913, 117649, F49, 5) (dual of [117649, 117636, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(4916, 117652, F49, 6) (dual of [117652, 117636, 7]-code), using
- linear OA(732, 235305, F7, 5) (dual of [235305, 235273, 6]-code), using Gilbert–Varšamov bound and bm = 732 > Vbs−1(k−1) = 165539 315826 738151 492801 [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(732, 235304, F7, 6) (dual of [235304, 235272, 7]-code), using
- construction X with Varšamov bound [i] based on
(27, 27+6, large)-Net in Base 7 — Upper bound on s
There is no (27, 33, large)-net in base 7, because
- 4 times m-reduction [i] would yield (27, 29, large)-net in base 7, but