Best Known (21, 21+5, s)-Nets in Base 7
(21, 21+5, 117651)-Net over F7 — Constructive and digital
Digital (21, 26, 117651)-net over F7, using
- net defined by OOA [i] based on linear OOA(726, 117651, F7, 5, 5) (dual of [(117651, 5), 588229, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(726, 235303, F7, 5) (dual of [235303, 235277, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(726, 235304, F7, 5) (dual of [235304, 235278, 6]-code), using
- trace code [i] based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 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(4910, 117649, F49, 4) (dual of [117649, 117639, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(726, 235304, F7, 5) (dual of [235304, 235278, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(726, 235303, F7, 5) (dual of [235303, 235277, 6]-code), using
(21, 21+5, 235304)-Net over F7 — Digital
Digital (21, 26, 235304)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(726, 235304, F7, 5) (dual of [235304, 235278, 6]-code), using
- trace code [i] based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 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(4910, 117649, F49, 4) (dual of [117649, 117639, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
(21, 21+5, large)-Net in Base 7 — Upper bound on s
There is no (21, 26, large)-net in base 7, because
- 3 times m-reduction [i] would yield (21, 23, large)-net in base 7, but