Best Known (61, 61+13, s)-Nets in Base 7
(61, 61+13, 39217)-Net over F7 — Constructive and digital
Digital (61, 74, 39217)-net over F7, using
- net defined by OOA [i] based on linear OOA(774, 39217, F7, 13, 13) (dual of [(39217, 13), 509747, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(774, 235303, F7, 13) (dual of [235303, 235229, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 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(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(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(12) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(774, 235303, F7, 13) (dual of [235303, 235229, 14]-code), using
(61, 61+13, 235304)-Net over F7 — Digital
Digital (61, 74, 235304)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(774, 235304, F7, 13) (dual of [235304, 235230, 14]-code), using
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 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(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(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(12) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
(61, 61+13, large)-Net in Base 7 — Upper bound on s
There is no (61, 74, large)-net in base 7, because
- 11 times m-reduction [i] would yield (61, 63, large)-net in base 7, but