Best Known (54, 54+12, s)-Nets in Base 7
(54, 54+12, 19613)-Net over F7 — Constructive and digital
Digital (54, 66, 19613)-net over F7, using
- net defined by OOA [i] based on linear OOA(766, 19613, F7, 12, 12) (dual of [(19613, 12), 235290, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(766, 117678, F7, 12) (dual of [117678, 117612, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(765, 117677, F7, 12) (dual of [117677, 117612, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(765, 117677, F7, 12) (dual of [117677, 117612, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(766, 117678, F7, 12) (dual of [117678, 117612, 13]-code), using
(54, 54+12, 117679)-Net over F7 — Digital
Digital (54, 66, 117679)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(766, 117679, F7, 12) (dual of [117679, 117613, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(765, 117677, F7, 12) (dual of [117677, 117612, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(765, 117678, F7, 11) (dual of [117678, 117613, 12]-code), using Gilbert–Varšamov bound and bm = 765 > Vbs−1(k−1) = 8482 139764 124177 780359 938517 547708 368407 066357 277159 [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(765, 117677, F7, 12) (dual of [117677, 117612, 13]-code), using
- construction X with Varšamov bound [i] based on
(54, 54+12, large)-Net in Base 7 — Upper bound on s
There is no (54, 66, large)-net in base 7, because
- 10 times m-reduction [i] would yield (54, 56, large)-net in base 7, but