Best Known (87, 105, s)-Nets in Base 7
(87, 105, 26145)-Net over F7 — Constructive and digital
Digital (87, 105, 26145)-net over F7, using
- net defined by OOA [i] based on linear OOA(7105, 26145, F7, 18, 18) (dual of [(26145, 18), 470505, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(7105, 235305, F7, 18) (dual of [235305, 235200, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(7104, 235304, F7, 18) (dual of [235304, 235200, 19]-code), using
- trace code [i] based on linear OA(4952, 117652, F49, 18) (dual of [117652, 117600, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4949, 117649, F49, 17) (dual of [117649, 117600, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(4952, 117652, F49, 18) (dual of [117652, 117600, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(7104, 235304, F7, 18) (dual of [235304, 235200, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(7105, 235305, F7, 18) (dual of [235305, 235200, 19]-code), using
(87, 105, 235306)-Net over F7 — Digital
Digital (87, 105, 235306)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7105, 235306, F7, 18) (dual of [235306, 235201, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(7104, 235304, F7, 18) (dual of [235304, 235200, 19]-code), using
- trace code [i] based on linear OA(4952, 117652, F49, 18) (dual of [117652, 117600, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4949, 117649, F49, 17) (dual of [117649, 117600, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(4952, 117652, F49, 18) (dual of [117652, 117600, 19]-code), using
- linear OA(7104, 235305, F7, 17) (dual of [235305, 235201, 18]-code), using Gilbert–Varšamov bound and bm = 7104 > Vbs−1(k−1) = 11 902900 623162 456076 864330 667369 564827 303587 618199 301069 651924 415907 728805 908142 610881 [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(7104, 235304, F7, 18) (dual of [235304, 235200, 19]-code), using
- construction X with Varšamov bound [i] based on
(87, 105, large)-Net in Base 7 — Upper bound on s
There is no (87, 105, large)-net in base 7, because
- 16 times m-reduction [i] would yield (87, 89, large)-net in base 7, but