Best Known (68−12, 68, s)-Nets in Base 7
(68−12, 68, 39217)-Net over F7 — Constructive and digital
Digital (56, 68, 39217)-net over F7, using
- net defined by OOA [i] based on linear OOA(768, 39217, F7, 12, 12) (dual of [(39217, 12), 470536, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(768, 235302, F7, 12) (dual of [235302, 235234, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(768, 235304, F7, 12) (dual of [235304, 235236, 13]-code), using
- trace code [i] based on linear OA(4934, 117652, F49, 12) (dual of [117652, 117618, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4931, 117649, F49, 11) (dual of [117649, 117618, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(4934, 117652, F49, 12) (dual of [117652, 117618, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(768, 235304, F7, 12) (dual of [235304, 235236, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(768, 235302, F7, 12) (dual of [235302, 235234, 13]-code), using
(68−12, 68, 235304)-Net over F7 — Digital
Digital (56, 68, 235304)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(768, 235304, F7, 12) (dual of [235304, 235236, 13]-code), using
- trace code [i] based on linear OA(4934, 117652, F49, 12) (dual of [117652, 117618, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4931, 117649, F49, 11) (dual of [117649, 117618, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(4934, 117652, F49, 12) (dual of [117652, 117618, 13]-code), using
(68−12, 68, large)-Net in Base 7 — Upper bound on s
There is no (56, 68, large)-net in base 7, because
- 10 times m-reduction [i] would yield (56, 58, large)-net in base 7, but